CONTROL OF INFINITE DIMENSIONAL SYSTEMS USING FINITE
DIMENSIONAL TECHNIQUES: A SYSTEMATIC APPROACH
by
Armando Antonio Rodriguez
B.S., Electrical Engineering, Polytechnic Institute of New York
(1983)
S.M., Electrical Engineering, Massachusetts Institute of Technology
(1987)
E.E., Electrical Engineering, Massachusetts Institute of Techmology
(1989)
Submitted to the Department of
ELECTRICAL ENGINEERING AND COMPUTER SCIENCE
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
August 1990
@Armando Antonio Rodriguez, 1990
The author hereby grants permission to M.I.T to reproduce and to distribute copies of this thesis
document in whole or in part
Signature of Author:..............
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Department of Electrical Engineering anVComputer Science
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Certified by: ......................................
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Accepted by: ................................................
MASSACHUSETTS
INSTITUTE
OF TECHNOLOGY
NOV 27 1990
,ARARIES
ARCHIVES
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Arthur C. Smith, Chairman
Committee on Graduate Students
Control of Infinite Dimensional Systems using Finite Dimensional Techniques: A
Systematic Approach
by: Armando Antonio Rodriguez
Submitted to the Department of Electrical Engineering and Computer Science on August 15, 1990
in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
ABSTRACT
In this thesis, the problem of designing finite dimensional controllers for infinite dimensional
single-input single-output systems is addressed. More specifically, it is shown how to systematically
obtain near-optimal finite dimensional compensators for a large class of scalar infinite dimensional
plants. The criteria used to determine optimality are standard H** and ?X 2 weighted sensitivity
and mixed-sensitivity measures.
Unlike other approaches which appear in the literature, the approach taken here avoids solving an infinite dimensional optimization problem to get an infinite dimensional compensator and
then approximating to get an appropriate finite dimensional compensator. Rather than this Design/Approximate approach, we take an Approximate/Designapproach. In this approach one starts
with a "good" finite dimensional approximant for the infinite dimensional plant and then solves a
finite dimensional optimization problem to get a suitable finite dimensional compensator. Traditionally, however, this approach has not come with any guarantees.
The key difficulties which have arisen can be attributed to the fact that these measures are
sometimes not continuous with respect to plant perturbations, even when the uniform topology is
imposed. Moreover, even if they were, it is a known fact that many interesting infinite dimensional
plants can not be approximated in the uniform topology on -Oo(e.g. a delay). Also, it must be
noted that the concept of a "good" approximant, in the context of feedback design, has never been
rigorously formulated.
The goal and main contribution of this research endeavour has been to resolve these difficulties.
It is shown that given a "suitable" finite dimensional approximant for an infinite dimensional
plant, one can solve a "natural" finite dimensional problem in order to obtain a near-optimal finite
dimensional compensator. Moreover, very weak conditions are presented to indicate what a "good"
approximant is.
In addition, we show that the optimal performance for a large class of W11o design paradigms can
be computed by solving a sequence of finite dimensional eigenvalue/eigenvector problems rather
than the typical infinite dimensional eigenvalue/eigenfunction problems which appear in the literature. Analogous results are presented for a large class of H2 paradigms.
In summary, the approach taken here allows one to forgo solving "complex" infinite dimensional
problems and provides rigorous justification for some of the approximations that control engineers
typically make in practice.
Thesis Supervisor: Dr. Munther A. Dahleh
Title: Professor of Electrical Engineering & Computer Science
Acknowlegdements
First and foremost, I would like to thank my thesis supervisor, Professor Munther Dahleh. He
has always offered infinite support, both technical and otherwise. His guidance, patience, and deep
concern have been very much appreciated. I have been extremely fortunate to have had him as my
thesis supervisor. I cannot thank him enough. I feel lucky not only to have had the opportunity
to learn from him, as a student, but also to have associated with him as a person. Although I will
see him at conferences in the future, I will still miss him a great deal.
I would also like to thank my thesis committee members Dr. Gunter Stein and Professor Sanjoy
Mitter for their helpful consultation throughout the course of this research. Every session with
them provided perspective and great insight. It was a pleasure and an honor to work with them.
In addittion, I must thank Professor Michael Athans, Ignacio Diaz-Bobillo, Joel Douglas, Professor David Flamm, Dr. Petros Kapasouris, Sanjeev Kulkarni, Joseph Lutsky, Dr. Dragan Obradovid,
Professor Jason Papastavrou, Dr. Tom Richardson, Professor Brett Ridgely, Professor Jeff Shamma,
Petros Voulgaris, Jim Walton. All have contributed in some way or form.
The staff at LIDS was always helpful and supportive. I would particularly like to thank Fifa
Monserrate. She was always kind, warm, and cheerful.
I must also give great thanks to AT&T Bell Laboratories for their most generous fellowship.
I would like to specifically thank my mentor at the labs, Dr. Sid Ahuja. He has always been
supportive.
Finally, I must give special thanks to my father and my brothers David and Raul for the love,
encouragement, and support that they have given me over the years. I would also like to thank
my dear friend Ted Hernandez who unfortunately passed away during this last year. I owe him so
much.
This research was conducted at the M.I.T Laboratory for Information and Decision Systems.
It has been supported by AT&T Bell Laboratories, by the Center for Intelligent Control Systems
under the Army Research Office grant DAAL03-86-K-0171, and by NSF grant 8810178-ECS.
To my father.
ol
Contents
1 Introduction & Overview
1.1 M otivation . . . . . . . . . . . . . .
1.2 Related Work & Previous Literature
1.3 Contributions of Thesis . . . . . . .
1.4 Organization of Thesis . . . . . . . .
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Mathematical Preliminaries: Notation & Function
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
2.2 Some Notation & Definitions . . . . . . .
2.3 Single-Valued Complex Functions . . . . .
2.4 Multi-Valued Complex Functions . . . . . . . . . . .
2.5 L2 Function Theory . . . . . . . . . . . . . . . . . .
2.6 Cw Function Theory . . . . . . . . . . . . . . . . . .
2.7 H** Function Theory . . . . . . . . . . . . . . . . . .
2.8 Theory of Linear Operators . . . . . . . . . . . . . .
2.9 Hankel/Toeplitz Operator Theory . . . . . . . . . .
2.10 N* Approximation Theory . . . . . . . . . . . . . .
2.11 Duality Theory . . . . . . . . . . . . . . . . . . . . .
2.12 Equicontinuity, Normality, Arzela-Ascoli .
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2.13 Summary . . . . . . . . . . . . . . . . . . . . . . . .
3 Results from Algebraic Systems Theory
3.1 Introduction . . . . . . . . . . . . . . . .
3.2 The Youla et al. Parameterization . . .
3.3 A Stability Result . . . . . . . . . . . .
3.4 The Corona Theorem . . . . . . . . . .
3.5 Summary . . . . . . . . . . . . . . . . .
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Statement of Fundamental Problems
4.1 Introduction . . . . . . . . . . . . . . . . . . . .
4.2 Basic Assumptions & Notation . . . . . . . . .
4.3 A/-Norm Approximate/Design J-Problem . . .
4.4 K-Norm Purely Finite Dimensional J-Problem
4.5 K-Norm Loop Convergence J-Problem . . . . .
4.6 Summary . . . . . . . . . . . . . . . . . . . . .
Theory
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Model Matching Problem
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Infinite Dimensional R** Model Matching Problem . . . . . . .
Computation of p . . . . . . . . . . . . . . . . . . . . . . . . .
Computation of Hankel Norm . . . . . . . . . . . . . . . . . . .
Sequences of Finite Dimensional R** Model Matching Problems
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6 Design via R" Sensitivity Optimization
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 R** Approximate/Design Sensitivity Problem . . . . . . . . . . .
6.3 Why is the Approximate/Design Problem Hard? . . . . . . . . .
6.4 Solution to ** Approximate/Design Sensitivity Problem . . . .
6.5 Solution to 7i* Purely Finite Dimensional Sensitivity Problem:
Optimal Performance . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Solution to ?** Loop Convergence Sensitivity Problem . . . . . .
6.7 Summ ary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Design via H** Mixed-Sensitivity Optimization
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7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.2 H** Approximate/Design Mixed-Sensitivity Problem . . . . . . . . . . . . . . . . . . 104
7.3
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7.4
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Solution to R1** Approximate/Design Mixed-Sensitivity Problem . .
Solution to IY** Purely Finite Dimensional Mixed-Sensitivity Problem:
of Optimal Performace . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6 Poles and Zeros on the Imaginary Axis . . . . . . . . . . . . . . . . .
7.7 Solution to hoo Loop Convergence Mixed-Sensitivity Problem . . . .
7.8 Unstable Plants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.9 General Weighting Functions . . . . . . . . . . . . . . . . . . . . . .
7.10 Super-Optimal Performance Criteria . . . . . . . . . . . . . . . . . .
7.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Design via 7 2 Optimization
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 R 2 Approximate/Design Sensitivity Problem . . . . . . . . . . .
8.3 Why is the Approximate/Design Problem Hard? . . . . . . . . .
8.4 Solution to H2 Approximate/Design Sensitivity Problem . . . . .
8.5 Solution to 7 2 Purely Finite Dimensional Sensitivity Problem:
Optimal Performance . . . . . . . . . . . . . . . . . . . . . . . . .
8.6 Solution to R2 Loop Convergence Sensitivity Problem . . . . . .
8.7 Unstable Plants . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.8 Mixed-Sensitivity and Super-Optimal Performance Criteria . . .
8.9 Sum mary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Summary and Directions for Future Research
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9.2 Directions for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
List of Figures
3.1
Feedback Control System Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
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Visualization of Approximate/Design Methodology . . . . . . . . . . . . . . . . . . . 53
Chapter 1
Introduction & Overview
1.1
Motivation
During the 1980's, the need to address the control of large scale flexible structures has increased
immensely. Although this need has arisen primarily from SDI and aerospace applications, it has also
been fueled by complexity issues in power distribution and other areas. Because of these driving
forces, the problem of designing feedback control systems for infinite dimensional systems has
received considerable attention during the past decade. Researchers have endeavoured to develop
systematic design procedures. In this thesis such systematic design procedures are presented for
a large class of infinite dimensional systems. More specifically, it is shown how to obtain nearoptimal finite dimensional compensators for H 0 and H2 weighted sensitivity and mixed-sensitivity
performance criteria. The approach taken in this thesis is now motivated.
Throughout the thesis, it shall be assumed that the designer has been given a single-input
single-output infinite dimensional plant' and a performance measure. In the spirit of the seminal work of Zames [59], it will also be assumed that the performance measure has been posed
as an infinite dimensional optimization problem. The goal then is to design a near-optimal finite
dimensional compensator. The finite dimensionality, of course, is a typical "real-world" implementation constraint. This thesis addresses the problem of designing near-optimal finite dimensional
compensators. Two approaches to this problem have appeared in the literature.
The first approach we call the Design/Approzimate approach. In this approach an optimal infinite dimensional compensator is designed by solving, if possible, the infinite dimensional optimization problem. The optimal compensator is then approximated by a finite dimensional compensator.
This approach will not be considered in the sequel.
The second approach we call the Approximate/Design approach. In this approach the infinite
dimensional plant is approximated by a sequence of finite dimensional plants. We then solve a
sequence of "natural" finite dimensional problems in which we simply substitute the finite dimensional approximants for the infinite dimensional plant in the original optimization problem. This
sequence of finite dimensional problems generate a sequence of finite dimensional compensators
which, ideally, will be near-optimal as the plant approximants get "better". Traditionally, however,
no such guarantees have been shown.
The key difficulties which have arisen can be attributed to the fact that these performance
measures are often not continuous with respect to plant perturbations, even when the uniform
topology is imposed. In this thesis, these difficulties are resolved; guarantees are provided.
'The system to be controlled is called the plant.
-I
_M9
12=
-1-1-
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The primary motive behind the Approximate/Design approach taken in this work has been that
of finding near-optimal finite dimensional compensators for scalar infinite dimensional systems.
Other motives can be listed as follows.
(1) Some infinite dimensional models are too complex. It is often very difficult to gain intuition
from them. Designing controllers based on such models often requires advanced mathematical
machinery and new software. It follows naturally to ask: What would be a "good" finite dimensional
approximant? Such an approximant should give immediate insight. To design controllers which are
based on such an approximant usually requires little mathematical sophistication. Moreover, much
software exists for such a finite dimensional approach. The above question raises the following
question: What information about the infinite dimensional plant do we really need in order to
achieve the control objective? The approach taken in this thesis attempts to shed light on the
above questions.
(2) Some design procedures result in compensators which are infinite dimensional. Such compensators may be difficult, if not impossible to implement. The following natural question thus arises:
How can we obtain a finite dimensional compensator which is suitable? The Approximate/Design
approach taken in this thesis addresses this question directly.
(3) Often, in the early stages of system planning and design, it is necessary to estimate achievable
system performance. Such an estimate could be used for system reconfiguration and enhancement.
It thus follows that efficient computational tools to obtain such performance information would be
extremely valuable to system designers. By taking an Approximate/Design approach, one addresses
such computational issues indirectly.
1.2
Related Work & Previous Literature
The problem of designing compensators for infinite dimensional plants has recieved considerable
attention during the past decade. Some relevant works are [1], [6]-[10], [13]-[20], [25]-[26], [31]-[38],
[40]-[41], [45], [47], [51]-[53], [57]-[63]. We now give a chronological summary of some of this work.
1950's
The works of [9], [10], and [32] address approximation issues. Real-rational approximation
methods are presented. The methods are based on "open loop" ideas and not on closed loop performance criteria.
1960's
Much of the technical issues which arise in todays X** model matching approach to control
synthesis, were addressed in the famous paper of [51]. In this paper the author solves various
interpolation problems in 7Y**. The paper contains the commutant lifting theorem and shows how
one can construct norm preserving R" dilations for various operators. This work has been the
cornerstone of many approaches/solutions which have appeared for the X** sensitivity and mixedsensitivity problems.
1970's
The Hankel matrix approximation problem is solved in [1]. This paper has also tremendously
influenced the 'H* model matching approach which is present everywhere in the control literature.
At the heart of todays model matching approach to control is the ability to parameterize all
internally stabilizing compensators for a given plant. Such a parameterization was done in [58] for
finite dimensional multivariable linear time invariant plants.
An algebra of transfer functions for distributed linear time invariant systems is presented in
[4]. Elements in the fraction field of this algebra possess coprime factorizations over the algebra.
The work of [58] can thus be used to parameterize the set of all internally stabilizing controllers for
plants which lie within the fraction field of the algebra.
1981
In [59], the author posed what is now referred to as the weighted 1X"sensitivity control problem.
One objective of this seminal paper was to formulate control problems as optimization problems
in an attempt to systematize control system design. It was argued and shown that this frequency
domain approach is natural to handle unstructured uncertainty.
1982
The work of [21] gives a very nice solution for scalar weighted W 2 sensitivity and mixedsensitivity problems.
1983
In [60], the authors use duality and interpolation theory to solve the weighted V"o sensitivity
problem for real-rational scalar plants. A fundamental motive for this work was to replace the
heuristic aspects of classical design by an explicit mathematical theory.
1984
The commutant lifting theory of [51] is used in [22] to obtain an upperbound for the optimal
weighted R" sensitivity associated with scalar finite dimensional systems. The problem of achieving
a small sensitivity over a specified frequency band is also addressed. The effects of non-minimum
phase zeros is discussed.
A solution to the scalar weighted XV0 mixed-sensitivity problem is presented in [55]. Here the
mixed-sensitivity criterion used is that which penalizes the sensitivity and the complementary sensitivity transfer functions.
1985
Necessary and sufficient conditions for the existence of finite dimensional compensators for delay
systems are presented in [31]. Moreover, it is shown that a stabilizable delay system can always be
stabilized using a finite-dimensional compensator.
1986
A method for constructing finite dimensional compensators which stabilize infinite dimensional
systems with unbounded input operators is presented in [6]. Applications to retarded and partial
differential equations are considered.
In [7], the authors present a method for contructing robustly stabilizing finite dimensional
compensators for a class of infinite dimensional plants.
The commutant lifting ideas in [51] are used by [14] to solve the weighted 1" sensitivity problem for the case where the plant is a product of a delay and a real-rational scalar function. Fairly
general real-rational weighting functions are considered. The optimal sensitivity and compensator
are computed in various situations. The implementation of the optimal infinite dimensional compensator is also discussed.
In [18], the authors also solve a weighted R" sensitivity problem using the work of [51]. Here
the plant is a delay and the weighting function is first order and strictly proper.
m
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1987
The ideas presented in [14] are expanded upon in [15].
The results of [18] are extended in [19] to more general delay systems. The plant is assumed
to be the product of a delay and a scalar real-rational transfer function with no poles or zeros on
the imaginary axis. The weight is assumed to be an RRIX" function which is invertible in X"O . The
interaction between delays and non-minimum phase zeros is also discussed.
A solution to the weighted RO" sensitivity problem is presented in [20] for arbitrary scalar
distributed plants. An explicit formula is given for the optimal sensitivity. The existence and
uniqueness of the optimal compensator is discussed.
In [63], the authors show how the optimal weighted X 0 sensitivity for a delay can be computed
by solving a two point boundary value problem. Here the weight can be any RX"2' function.
The uniform approximation of a class of delay systems by means of partial fraction expansions
is investigated in [62]. Nuclear systems are discussed.
1988
Krein space theory is used to tackle the X" mixed-sensitivity problem in [13]. Here the mixedsensitivity criterion considered penalizes the sensitivity and complementary sensitivity functions.
In [16], the authors expand upon the implementation issues presented in [14].
The problem of uniformly approximating delay systems is considered in [45]. Condition for
nuclearity are given.
In [25], the authors show how to construct real-rational approximants for nuclear systems. More
specifically, it is shown that for this class of systems, balanced or output normal realizations always
exist and their truencations converge to the original system in various topologies. Various error
bounds are given.
An iterative procedure for constructing near-optimal infinite dimensional compensators for a
class of infinite dimensional plants is presented in [57]. The optimality criterion is an R1-*sensitivity
criterion. The method presented assumes that the weighting function is strictly proper. In such a
case the corresponding Hankel operator is compact.
The computation of the essential spectra of certain Hankel-Toeplitz operator pairs is crucial
in the solution of H** sensitivity and mixed-sensitivity problems. This is particularly important
when infinite dimensional systems are involved. Such a computation is given in [61]. Calkin algebra
techniques are used to obtain the results.
1989
In [8], the author addresses the control of infinite dimensional systems which belong to the
algebra presented in [4]. Systems which are of the Pritchard-Salamon class are also addressed.
In [12], the authors present state space formulae to solve a myriad of finite dimensional X 2 and
w* control problems.
An FFT-based algorithm for approximating infinite dimensional systems is presented in [28].
When approximating infinite dimensional systems by finite dimensional approximants, the rate
at which the approximants converge is very important. Such convergence rate results are given in
[26] for certain approximants and infinite dimensional systems. Pade approximations of delays, for
example, are discussed.
An H** mixed-sensitivity problem for a flexible Euler-Bernoulli beam is solved in [34]. The
solution relies on the techniques used in [41].
In [36], the author uses Laguerre series to approximate certain infinite dimensional systems.
Various error bounds are given.
I
~-'~~--
Skew Toeplitz techniques are used in [40], to solve various control problems for infinite dimensional systems. The V** mixed-sensitivity problem is addressed.
In [41], the authors reveal the structure of suboptimal X* controllers for distributed plants.
In [52], the author shows that fraction field of X** is a Bezout domain. That implies that
plants in the fraction field of *' possess coprime factorizations over 7**. This, then allows us to
paramerize all stabilizing compensators for such plants using the ideas of [58].
1990
A solution to the ?1 mixed-sensitivity problem is presented in [17]. Here, very general distributed plants and irrational weighting functions are treated.
7** mixed-sensitivity techniques are used in [42] to control unstable infinite dimensional plants.
Skew Toeplitz methods are used to derive the controllers.
In [53], the author studies the continuity properties of various XP problems. In particular, it
was shown that 1* and N 2 problems, in general, are discontinuous functions of the plant, even
when the N** topology is used. This paper contains many of the ideas presented in this thesis. It
does not, however, address control design. It only addresses certain well-possedness issues which
arise in certain optimization problems. We also would like to point out that although elements
of this work was initially published in 1987, in a conference proceedings, it did not come to our
attention until March 1990.
1.3
Contributions of Thesis
In this thesis, the Approximate/Design Problemis rigorously formulated. A solution is provided for
N** and N 2 weighted sensitivity and mixed-sensitivity performance criteria. These are the main
contributions of the thesis.
More specifically, it is shown that given a "good" finite dimensional approximant for an infinite
dimensional plant, one can solve a "natural" finite dimensional problem in order to obtain a nearoptimal finite dimensional compensator. Conditions are given which precisely quantify the notion
of a "good" finite dimensional approximant.
Given this, the contributions of the thesis can be concretely stated as follows.
(1) A method for constructing near-optimal finite dimensional controllers for a large class of
infinite dimensional scalar plants is presented. Stable and unstable plants can be handled. Much
software exists to support the necessary computations. The method is thus immediately implementable by practicing engineers.
(2) The same method can be used to construct near-optimal infinite dimensional controllers.
One can then "directly" obtain near-optimal finite dimensional compensators by approximating
the infinite dimensional controllers. This approach, however, goes against the spirit of the thesis
since it requires that we solve an infinite dimensional optimization problem.
(3) The most commonly used design criteria have been addressed; i.e. N"* and N 2 weighted
sensitivity and mixed-sensitivity problems. Again, much software exists to support our finite dimensional approach to these paradigms.
(4) For a large class of N"* weighted sensitivity/mixed-sensitivity problems, the optimal performance can be easily computed by solving a sequence of finite dimensional eigenvalue/eigenvector
problems rather than the typical infinite dimensional eigenvalue/eigenfunction problems which
appear in the literature. The methods presented apply even in situations where the associated
Hankel/Toeplitz operators are non-compact.
Such information can be used to determine fundamental performance limitations for a given
system; e.g. best possible £2 disturbance rejection, robustness, etc. It can thus be used to guide
designers (engineers, pilots, etc.) during the initial stages of system development, design, and
configuration.
Analogous results are presented for the X 2 design criteria considered.
(5) The thesis sheds light on such issues as what a "good" finite dimensional approximant is
and hence on what information is needed in order to achieve a particular control objective. It is
shown that appproximations should be based on the control objective; not on open loop intuition.
Moreover, it is shown that open loop intuition can often be quite misleading. These ideas have
potential implications in such fields as system identification and decentralized control.
1.4
Organization of Thesis
The remainder of this thesis is organized as follows.
In Chapter 2, notation and results from complex variable and approximation theory are presented. The function spaces H** and R 2 are defined and discussed. Various notions of convergence
are presented.
In Chapter 3, results from algebraic system theory are presented; e.g. the Youla parameterization and the Corona theorem.
In Chapter 4 three problems are formulated. They are the A-Norm Approximate/Design JProblem, the K-Norm Purely Finite Dimensional J-Problem, and the K-Norm Loop Convergence
J-Problem. In the sequel, the K will represent HX** and X 2 norms. The J will represent sensitivity
and mixed-sensitivity performance criteria.
The K-Norm Approximate/Design J-Problem addresses the problem of finding near-optimal
finite dimensional compensators. The K-Norm Purely Finite dimensional J-Problem addresses
the issue of computing the optimal performance using finite dimensional techniques. The K-Norm
Loop Convergence J-Problem addresses the question of what additional properties are exhibited
by designs based on the Approximate/Design approach advocated in the thesis. The above three
problems are considered in the sequel for Ro* and R 2 weighted sensitivity and mixed-sensitivity
performance criteria.
In Chapter 5, the H** Model Matching Problem is defined and discussed. It is shown how
near-optimal solutions can be constructed. The results presented here are exploited heavily in
subsequent chapters on R 0* design.
In Chapter 6, the focus is on designing near-optimal finite dimensional compensators based on
00
* sensitivity design criteria. More specifically, in this chapter a solution is presented to the R**
Approximate/Design Sensitivity Problem, the H 0 Purely Finite Dimensional Sensitivity Problem,
and the 7** Loop Convergence Sensitivity Problem. For simplicity, stable and unstable plants are
treated separately.
In Chapter 7, the focus is on designing near-optimal finite dimensional compensators based
on R 0 mixed-sensitivity design criteria. More specifically, in this chapter a solution is presented
to the N** Approximate/Design Mixed-Sensitivity Problem, the R 0 Purely Finite Dimensional
Mixed-Sensitivity Problem, and the N00 Loop Convergence Mixed-Sensitivity Problem.
In Chapter 8, the focus is on designing near-optimal finite dimensional compensators based
ffiWjjAS=jN---
-
on
-,--
R2
-
-
sensitivity/mixed-sensitivity design criteria. More specifically, in this chapter a solution is
presented to the H2 Approximate/Design Sensitivity Problem, the H 2 Purely Finite Dimensional
Sensitivity Problem, and the H2 Loop Convergence Sensitivity Problem. The analogous mixed-
sensitivity problems are also addressed. The features which distinguish the X2 case from the R**
case are highlighted.
Finally, Chapter 9 summarizes the results of the thesis and suggests possible directions for
future research.
Chapter 2
Mathematical Preliminaries:
Notation & Function Theory
2.1
Introduction
In this chapter we establish notation to be used throughout the thesis. Some essential mathematical
results are also presented. More specifically, the normed linear spaces H 2 and Roo are defined
and discussed. Various notions of convergence in HX1" are presented; e.g. uniform and compact
convergence. Results from hX-"* approximation theory are also presented. Examples are given to
illustrate some of the ideas. Most of the material presented in this chapter can be found in [2], [5],
[30], [33], [35], [44], [49], [56] 1.
2.2
Some Notation & Definitions
Throughout the thesis, we will use the symbols C, R, and Z to denote the complex, real, and
integer numbers, respectively. C, and Re will be used to denote the extended complex and real
numbers. The open right and left half complex planes will be denoted C+ and C_. R+ and Z+
will be used to denote the non-negative real numbers and positive integers. I()| and (.) will denote
the magnitude (modulus) and complex conjugate of the complex quantity (-). () and L(.) will be
used to denote the phase angle of the complex quantity (.). The symbol j will be used to denote
the purely imaginary number VT.
The greek letter e shall always be used in proofs to denote a given strictly positive, but arbitrarily
small, quantity. With this convention we can avoid the excess verbage "given e > 0, however
small,...".
Given sets A and B, A C B and A C B will be used to denote strict and non-strict containment
of A within B. The symbol A/B will denote the set of points in A which are not in B. A U B and
A n B will denote the union and intersection of the sets, respectively.
In the mathematics literature the characteristicfunction of a set S is defined as that function
which is unity on S and zero elsewhere. It is typically denoted XS. Throughout the thesis we
shall predominantly work on the imaginary axis. This motivates the following definition which we
introduce strictly for notational economy.
'Extra material has been included in this chapter in order to make the thesis self-contained and to facilitate future
addenduns.
Definition 2.2.1 (Characteristic Function)
Let S denote a subset of the extended real numbers. In what follows, the map Xs : jS -+ {0, 1}
will denote the characteristicfunction of the set jS; i.e.
Xs(jw)
cf
1
w E S;
0 elsewhere.
Convention 2.2.1 (Transform Pairs)
We shall often use f(t) or f and F(s) or F, to denote Laplace, Fourier,Planchereltransform pairs.
Here f (lowercase) will denote a time function and F (uppercase) will denote its transform. The
interpretation should be clear from the context.
Convention 2.2.2 (Lebesgue Integral)
All integrals in this thesis shall be assumed to be Lebesgue integrals [49], unless otherwise stated.
Any measure-theoretic statements which appear are made with respect to Lebesgue's measure
unless otherwise stated 2.
Let f(t) denote a real-valued Lebesgue measurable function defined on a set S C R.
Definition 2.2.2 (Support of a Function)
The support of f, denoted suppf, is defined to be the closure of the set where f takes on non-zero
values; i.e.
def
suppf = closure{t E S
I f(t) $
0}.
I
Definition 2.2.3 (Essential Supremum of a Function)
The essential supremum of f on S is defined as follows
ess sup f e inf{ M E Re I measure( {t E S I f(t) > M} ) = 0 }.
M
Here measure(.) denotes the Lebesgue measure of the set
(-).
U
Definition 2.2.4 (Convolution of Functions)
Given any two time functions f and g with support on R, their convolution will be denoted f * g
and defined as follows
(f
2
*
g)(t) jef
L
f(t -
r)g(r )dr.
Measure theoretic arguments shall be kept to a minimum throughout the thesis for added simplicity and brevity.
I
Throughout the thesis we will deal with normed linear spaces.
norm of a function f belonging to the normed linear space (.).
IIf II(.)
will be used to denote the
Definition 2.2.5 (Isometry, Isomorphism)
Let N1 and A2 be normed linear spaces. A norm preserving linear operator from Ar1 to A2 is called
an isometry. Such operators are necessarily injective (one-to-one). They need not be surjective
(onto). Ar1 and A2 are said to be isomorphic if there exists a bijective bounded linear operator
from Arl to A 2 whose inverse is also bounded (cf. definition 2.8.1). The operator is called an
isomorphism. A surjective isometry is an isomorphism. We call such an operator an isometric
isomorphism.
In what follows we shall deal with the normed linear spaces
shortly.
2.3
2 and N*. They shall be defined
Single-Valued Complex Functions
In this section we present standard definitions and results from the theory of complex functions of
a single complex variable. Let F(s) or F, denote a complex-valued function of a single complex
variable s. We assume all such functions to be single-valued unless otherwise stated. Let so denote
any point in the finite complex plane.
Definition 2.3.1 (Domain and Boundary)
A domain is a non-empty open connected subset of the extended complex plane. The boundary of
a domain V shall be denoted OD.
The open right half plane is a domain.
Definition 2.3.2 (Domain of Definition)
The domain of definition of a single-valued complex function is a domain in the complex plane over
which the function is defined.
In the sequel, the domain of definition of a function will be ascertainable from the context. It will
usually be the region of convergence of the function when viewed as the Laplace Transform of a
time function.
Definition 2.3.3 (Analytic and Entire Functions)
We say that F is analytic at the point so, if F is differentiable at all points in some open neighborhood of so. We say that F is analytic within a domain D, if it is analytic at each point within D.
If F is analytic within C, then we say that F is an entire function.
Some authors use the term holomorphic rather than analytic.
-
Proposition 2.3.1 (Taylor Series)
Let F be analytic at the point a = so. Given this, there exists a neighborhood
{s E C I Is - sol < R}
N(so, R)
about so, and a sequence {am}'o, such that
00
F(s) = E
am(s - so)"
m=O
for all s E N(so, R). Moreover,
am
=
F(m)(so)
and the series converges uniformly on all compact subsets within the neighborhood. We shall refer
to this representation for F as the Taylor series expansion for F about so.
U
The radius of convergence, of the Taylor series for F about so is equal to the distance from so to
the nearest singularity of F (cf. definition 2.3.7). It is given by Hadamard'sformula
Ro =I
limn+oo sup V/a 7
Definition 2.3.4 (Zero of a Function)
We say that so is a zero of F, if
lim F(s) = 0
8--+80
along any path in the domain of definition of the function F.
Definition 2.3.5 (Multiplicity of a Zero for Analytic Functions)
Let F be analytic at s = so. Also, let so be a zero of F. We say that so has multiplicity m E Z+,
if there exists a function G such that for all s in a neighborhood of so,
F(s) = (s - so) m G(s)
where G(so) j 0 and G is analytic within the neighborhood.
I
An integer m and a function G can always be found for any single-valued function F. In this
sense, all single-valued functions exhibit polynomial behavior near a zero. It must be noted that
for single-valued functions obtained from multi-valued functions, the situation is different. Such
functions shall be discussed in the next section.
Definition 2.3.6 (Roll-off)
We shall say that F rolls-off if oo is a zero of F.
I1
Definition 2.3.7 (Singularity, Isolated Singularity)
If F is not analytic at so, then we say that s = so is a singularityof F. If there exists a neighborhood
around so which contains no other singularities of F, then we say that so is an isolated singularity
of F.
It is standard convention by authors to assume that oo is a singularity. This convention shall receive
further consideration below. A zero at oo, we shall see, should be viewed as a removable singularity
(cf. definition 2.3.8).
Proposition 2.3.2 (Laurent Series)
Let so be an isolated singularity of F. Given this, there exists an annulus
{s E C I r1 < |s - so l < r 2)
A(so, ri, r 2 )
about so, and sequences {am}m*o, {bn}*
1
00
, such that
00
F(s) =
am(s - so)m + E bn
m=O
n=1
1
for all s E A(so, ri, r 2 ). Moreover, the convergence is uniform on all compact subsets of the annulus.
We shall refer to this representation for F as the Laurent expansion for F about so. The second
summation, which consists only of negative powers of (s - so), is called the principal or singular
part of F at so.
Let so be an isolated singularity of F.
Definition 2.3.8 (Removable Singularity)
The point s = so is said to be a removable singularity of F, if
lim (s - so)F(s) = 0
8-+30
along any path in the domain of definition of F [2, pp. 124].
One can show that the above is equivalent to F having no principal part. In such a case, the
function can be made analytic at so, simply by redefining it at the point.
One can also show that so is removable if and only if IF(s)I is bounded in some annulus about
SO. This condition is due to Riemann.
Definition 2.3.9 (Pole)
If lim,_.,, F(s) = oo, along any path in the domain of definition of F, then we say that the point
s = so is a pole of F.
One can show that this is equivalent to F having a principal part with only a finite number of
terms.
Definition 2.3.10 (Multiplicity of a Pole)
Let so be a pole of F. We say that it is a pole with multipliciity m, if so is a zero of " with
multiplicity m.
Definition 2.3.11 (Improper Function)
We say that F is improperif oo is a pole of F.
Definition 2.3.12 (Rational Function)
We say F is a rationalfunction, if it is the ratio of two polynomials; each of finite degree. We say
that it is real-rational,if the coefficients of the numerator and denominator polynomials are real.
U
Rational functions have a finite number of poles and zeros. They also possess singularities at o
which are removable.
Definition 2.3.13 (Meromorphic Function)
F is said to be meromorphic in a domain D if the only singularities within V are isolated poles. If
V = C, then we just say that the function F is meromorphic.
2-.
and G(s) =
are examples of a meromorphic functions.
The functions F(s) =
A meromorphic function can only have a finite number of poles in any compact set. If a
meromorphic function has an infinite number of poles, then they must cluster at oo.
It can be shown that a meromorphic function F with a finite number of poles is necessarily the
ratio of an entire function over a polynomial.
It can also be shown that meromorphic functions are necessarily the ratio of two entire functions.
Definition 2.3.14 (Essential Singularity)
We say that so is an essential singularityof F, if the principal part of its Laurent expansion about
so has an infinite number of nonzero terms.
One can show that this can occur if and only if so is a singularity of F which is neither removable
nor a pole.
Singularities at oo are treated as follows.
Definition 2.3.15 (Singularities at oo)
s = co is a removable singularity, a pole, or an essential singularity of F(s), if ( = 0 is a removable
singularity, a pole , or an essential singularity of FQ).
I
e- 8
The functions
and e have essential singularities at oo and 0, respectively.
The following proposition gives a remarkable characterization of essential singularities.
Proposition 2.3.3 (Picard's Theorem)
A complex-valued function F in any neighborhood of an essential singularity assumes all values
except possibly one.
This proposition is often referred to as Picard's Theorem. A weaker version of the theorem, known
as the Casorati-WeierstrassTheorem, appears in [33, pp. 158].
The above shows that isolated singularities of otherwise single-valued analytic functions must
either be removable singularities, poles, or essential essential singularities.
The following proposition characterizes "maxima" of analytic functions on a domain [2, pp.
134], [33, pp. 150], [49, pp. 212, 249, 253-259]. It is known as the Mazimum Modulus Theorem.
Proposition 2.3.4 (Maximum Modulus Theorem)
Let F be analytic within a domain D. Then IF(s)| achieves its maximum within D if and only if
F is constant. If the domain D is bounded, then we have
IF(s)| 5 sup IF(s)|
8EOD
for all s E D.
The above does not imply that an analytic function F on a domain will achieve its maximum on
the boundary. The following example illustrates this point.
Example 2.3.1 (Unbounded Analytic Function on a Strip)
The function F(s) = ee' is an entire function. Consider it over the unbounded domain jIm(s)|I
.
It does not achieve its maximum modulus on the boundary. Its magnitude on the boundary is 1.
Within the domain, however, the function grows exponentially along the positive real axis.
The following proposition can be found in [2, pp. 127], [49, pp. 209].
Proposition 2.3.5 (Uniqueness of Analytic Functions)
Let F and G be analytic within some domain D in the complex plane. Let S be a subset of D. If
S has an accumulation point in V and F = G on S then F = G throughout D.
I
This proposition implies that the zeros of a non-constant analytic function cannot accumulate
within its domain of analyticity. If they do, then the function must be identically zero over the
entire domain. This property is due to the fact that the zeros of a single-valued analytic function
are necessarily isolated within its domain of analyticity.
2.4
Multi-Valued Complex Functions
In this section we consider multi-valued complex functions of a single complex variable. Such
functions, as we shall see, can be viewed as a collection of single-valued functions. Let F denote a
complex valued function of a single complex variable s.
In order to precisely define multi-valued functions, we define the concept of a branch point as
follows. A branch point is a singularity that is associated with multi-valued functions only.
Definition 2.4.1 (Branch Point)
The point a = so is a branch point of the function F, if when we travel around the point, along a
sufficiently small circle, we do not return to the same value; i.e. for r > 0, sufficiently small,
2
F(so + r) / F(so + re' -).
Branch points at oo can be defined similarly by considering balls around the point at oo. Such a
ball could be precisely defined in terms of the Riemann sphere associated with the complex plane.
An equivalent approach is provided by the following definition. The definition parallels that given
for singularities of single-valued functions at oo in definition 2.3.15.
Definition 2.4.2 (Branch Points at oo)
The point s = oo is a branch point of the function F(s), if ( = 0 is a branch point of F().
I
Definition 2.4.3 (Multi-Valued Function)
F is a multi-valued function if it posseses a branch point in its domain of definition.
The function F(s) = v/s -1, for example, is multi-valued. It posseses a branch point at s = 1.
The analytic study of multi-valued functions usually requires that the multi-valued function be
expressed in terms of single-valued functions. One way of doing this is to consider the multi-valued
function in a restricted region of the extended complex plane. Then, one chooses a value at each
point in such a way that the resulting single-valued function is continuous on the restricted domain.
This motivates the following definition.
Definition 2.4.4 (Branch of a Multi-Valued Function)
A continuous single-valued function obtained from a multi-valued function is called a branch of the
multi-valued function.
Definition 2.4.5 (Branch Cut)
Typically, to obtain a branch, one must delete some curve in the s-plane. Such a curve is typically
referred to as a branch cut. The curve is such that if not crossed over, the function remains singlevalued. Given this, we have that a branch cut is a line or curve of singular points introduced in
defining a branch of a multi-valued function.
U
The branch cuts of a multi-valued function are not unique. A given branch cut, however, uniquely
defines a distinct branch of the multi-valued function. Branch points are conunon to all branch
cuts of a multi-valued function. A branch is also uniquely determined once a particular value of
the multi-valued function over the restricted domain has been specified. Specifying such a value
automatically determines the branch cut. This follows by continuity.
We now address multi-valued functions which shall be implicitly consider in the sequel.
Definition 2.4.6 (Complex Logarithm)
Let ln(.) denote the real-valued natural logarithm from elementary calculus. The complex logarithmic function is denoted Inm(.) and defined by the equation
def
nm
=(s)
IlnIs + jO(s)
where s = Islei(3) is any extended complex number and Inm,(O)
=f
-oo and In,,(oo) ' 00.
I
To see that this function is multi-valued we consider a circle of radius one centered at the origin
of the complex plane. We then note that if the circle is traversed in the conterclockwise direction,
then Inm,(.) does not return to its initial value. We have, for example, Inm,(le3) = 0 + jO and
Inmv(le12w) = 0 + j2r. The origin is thus a branch point of the complex logarithmic function.
Given this, we see that it is a multi-valued function. One can show that the point s = oo is also a
branch point. To obtain a branch of this function we proceed as follows.
Let
S def
= Ce/{-oo < Re(s) < 0; Im(s) = 0} = {s E C I - r < 6(s) <7r }
denote the set of points in the extended complex plane obtained by deleting the extended negative
real-axis. S defines a branch cut for the complex logarithmic function.
Definition 2.4.7 (Principal Angle)
We define the principalangle function O,, : S/0
(-r,
r) as follows
-tan-'
+
o < 0; o > 0;
- tan"
-
tan-1
6,,s) '
-+
W
0
7r
a
> 0;
<
0; W <
0
where s = a + jw.
We note that
,, is continuous on S/0.
Definition 2.4.8 (Principal Logarithm)
Given the above, the principal branchof the complex logarithmic function is defined on S as follows
ln,,(s)
for s
$
0, In,(0)
00 , and In,(oo)
def
= InIsI+ j0p(s)
de 00.
This function is analytic, as well as continuous, on S. The negative real-axis is the associated
branch cut. Other branch cuts and branches of the logarithmic function can similarly be defined.
Now we consider another multi-valued function. It is defined as follows.
Definition 2.4.9 (Complex Power Function)
Let c be a fixed real number. Given this, we have
(8C)Mv t
where s is any complex number.
ecln-,(s)
I
The principal branch of this function is defined by using the principal logarithm In,,(.) as
follows.
Definition 2.4.10 (Principal Power Function)
(S Z)P
4f
ez np (s)
Proposition 2.4.1 (Analytic Branches)
Given a multi-valued function, an analytic branch F can always be constructed in any domain
which does not contain a branch point.
U
This convention shall be adopted throughout the thesis for all multi-valued functions considered.
Given this, functions such as
F~s=
F)
1
3
2
s + 1
s+ 2
-lisj
s+3
4
s+4
will be regarded as analytic in the extended open right half plane and continuous everywhere on
the extended imaginary axis.
It was seen in the previous section that single-valued functions exhibit polynomial behavior
near zeros. This is not the case for single-valued functions which are constructed from multi-valued
functions. The function F(s) = fs, for example, exhibits irrational behavior near s = 0. We thus
have the following definition.
Definition 2.4.11 (Algebraic Multiplicity of a Zero)
Let F be a branch of a multi-valued function. Let F be analytic at s = so. Also, let so be a zero
of F. We say that so has algebraic multiplicity m E R+, if there exists a function G such that for
all s in a neighborhood of so,
F(s) = (s - so)m G(s)
where G(so) $ 0 and G is analytic within the neighborhood. Here the neighborhood is assumed to
lie within the domain of definition of the branch F.
A non-negative real nmnber m and a function G can always be found for any analytic branch F of
a multi-valued function.
2.5
£2
Function Theory
The function space L2 is defined as follows.
Definition 2.5.1 (Function Space: C2 (R))
,2 4f 1 2 (R) will denote the space of Lebesgue square integrable complex-valued functions with
support on R. C2 (R+) and £ 2 (R-) are similarly defined. C2 is a normed linear space over the field
C, when endowed with the following norm
||f||,C2
|f(t)|2dt.
The £2 functions should be thought of as the set of all finite energy time signals.
The space £2 is also an inner product space when endowed with the following inner product
< f, g >,C2
f0f(~tig(t)dt.
Moreover, it is complete with respect to this inner product. £2 is thus a Hilbert space.
The following proposition can be found in [35]. It is the basis for classical projection theory.
Proposition 2.5.1 (Classical Projection Theorem)
Let H denote a Hilbert space withe inner product
< -,. >,H. Let M denote a closed subspace of X.
Given x E ?Y there exists a unique element mo C M such that
min JJ - m||.
mEM
Moreover, x - m, E M'
= ||z - mJ||H.
where
M'
{x E -t i < x, m >%= 0 m E M}.
We say that m, is the projection of x onto M and write
Mo = fMX.
Also, R is the direct sum of calM and calM'; i.e.
This notation means that each element x c H can be written as X = m 1 + m 2 where mi E M and
m 2 E M'.
From the classical projection theorem, it thus follows that f 2 (R) is the direct sum of £ 2 (R+)
and C2 (R_):
=2(R)
= E 2 (R+) @£ 2(R);
f
f+ + f_.
i.e. given
f=
c E2 (R) there exists unique functions f+ E E2 (R+) and f- E 2 (R_), such that
This is because C2 (R_) is the orthogonal complement of E2 (R+) in £ 2 (R).
Definition 2.5.2 (Function Space: L2 (jR))
One can also define the space L 2 (jR) as the set of all complex-valued functions which are Fourier/Plancherel
transforms of £ 2 (R) time functions. £ 2 (jR) is a normed linear space over the field C when endowed
with the following norm
|FeC2ii~iic(
3 R00
1
|iF(jw) 2dw.
U
Parseval's theorem tells us that C2 (R) and E 2 (jR) are isomorphic. More specifically, it shows that
li|IC2(R) = ||F11c2(jR).
Since C2 (R) and C2 (jR) are isometrically isomorphic, there is no need to distinguish between
the spaces. We shall usually just write C2. Which space is being considered should be apparent
from the context.
The function space ?j 2 is defined as follows.
Definition 2.5.3 (Function Space: 7J2)
2 (C+)
72 4will denote the Hardy space of complex-valued functions which are analytic in C+
and uniformly Lebesgue square integrable on lines parallel to the imaginary axis in C+. ?y2 is also
a normed linear space over the field C, when endowed with the following norm
||Fllu 2 =Lsue -
|F(o-+ js)| 2dw.
consists exactly of those functions which are Laplace/Plancherel transforms of £2(R+) functions
[29, pp. 100, Paley-Wiener Theorem]. Consequently, there is no need to distinguish between R 2
and £ 2 (R+). They are isomorphic.
?X2
Definition 2.5.4 (Function Space: h2±)
Analogously, H2J will be used to denote the space of functions which are Laplace transforms of
functions in C 2 (R_).
It is a fact [30, pp. 128] that ?X2 and 2L functions can be unitarily extended to have support
almost everywhere on the imaginary axis. More precisely, we have the following proposition.
Proposition 2.5.2 (Extension of ?J2 Functions onto the Imaginary Axis)
Given F E ?j2, it possesses non-tangentiallimits almost everywhere on the imaginary axis. By
taking such limits, one obtains a unique extension P E £2 of F onto the imaginary axis. Moreover,
this extension is an isometry.
I
A similar proposition applies for functions in H2-. When dealing with R2 and R2 J functions
we shall always assume that we are dealing with the extended functions; i.e. we make no distinction
between the function and its extension. Given this, the norms of such functions can be computed
from values on the imaginary axis. This gives us the following proposition.
Proposition 2.5.3 (Norm of 12 Functions)
Given F E 1i 2 , we have
||F||K2 = ||F11c2 (IR).
I
The following notation shall be used to denote the various projection operators on C2 .
Definition 2.5.5 (Projection Operators on £2)
The projection of C2 onto L 2 (R+) ( or R2) shall be denoted llc2(R+) ( or Hu 2 ) . These operators
shall be used interchangeably. The projection of C2 onto £ 2 (R_) (or X2') shall be denoted HC2(R-)
(or R.
U).2
These operators shall be used interchangeably.
We note that such projection operators have induced norms, on
£2,
equal to 1.
LO Function Theory
2.6
The function space E** is defined as follows.
Definition 2.6.1 (Function Space: C**(jR))
,o w 4f**(jR) will denote the space of Lebesgue measurable essentially bounded complex-valued
functions with support on the imaginary axis. This space is a normed linear space over the field
C, when endowed with the following norm
||F||.
=
ess sup |F(jw)I.
wE Ri
Proposition 2.6.1 (System Interpretation)
Given F E VC*, the associated time function f defines a convolution kernel from
|IF||LM
where
f
* x denotes the convolution of
f
=
sup
£2
to £2. Moreover,
|if * X||C2
with x.
U
Definition 2.6.2 (Adjoint)
Given a complex-valued function F(s) with domain of definition including the imaginary axis, we
shall denote the adjoint of F as follows
F*(s) d F(-)
Strictly speaking, the concept of an adjoint should be defined in terms of bounded linear functionals
[48]. This degree of rigor will not be necessary for our purposes.
The function space C,(jR) is defined as follows.
Definition 2.6.3 (Function Space: Ce(jR) )
Ce df C,(jR) is the set of all complex-valued functions which are continuous on the extended
imaginary axis.
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2.7
'W** Function Theory
The function space W"' is defined as follows.
Definition 2.7.1 (Function Spaces: X", RO")
def
H 0=
7- (C+) will denote the Hardy space of complex-valued functions which are analytic and
essentially bounded in C+. 1"- is a normed linear space over the field C, when endowed with the
norm
|IFII% I sup sup jF(u + jw)I.
a>O WERe
RO
7"(C+) will denote the subspace of R" functions which roll-off.
RIX" and R1"1
will denote the corresponding subspaces of real-rational X" functions. We note
that Ri""functions have no poles in the extended closed right half plane [23]; i.e. all of their poles
lie in the open left half plane.
Definition 2.7.2 (Function Space: X(C_))
1"(C_) will denote the Hardy space of complex functions which are analytic and essentially
bounded in C.
I
RX" (C_) will denote the corresponding subspace of real-rational functions. Such functions have
no poles in the extended closed left half plane [23]; i.e. all of their poles lie in the open right half
plane.
Definition 2.7.3 ( 1 B)
1 B 4'f XB(C+) will be used to denote the Hardy space of complex-valued functions which are
analytic in C+ and bounded on compact subsets within C+ [60, pp. 589].
This space should be thought of as containing all improper functions which would be in 1" if it
were not for their improperness (e.g. f(s) = s). RRB will denote the corresponding subspace of
real-rational functions.
It is a fact [30, pp. 128] that 1"' functions can be unitarily extended to have support almost
everywhere on the imaginary axis. More precisely, we have the following proposition.
Proposition 2.7.1 (Extension of R" Functions onto the Imaginary Axis)
Given F E -", it possesses non-tangential limits ahnost everywhere on the imaginary axis. By
taking such limits, one obtains a unique extension F E L""of F onto the imaginary axis. Moreover,
this extension is an isometry.
I
When dealing with such functions we shall always assume that we are dealing with the extended
functions; i.e. we make no distinction between the function and its extension. Given this, the norms
of such functions can be computed from values on the imaginary axis. We state this formally as
follows.
Proposition 2.7.2 (Maximum Modulus Theorem for 7-**)
Given F E H**, we have
||FII~ioo = ||F||
ess sup IF(jw)|.
WER,
Consequently,
|F(sl}| <
F||g.
for all s such that Re(s) > 0.
Definition 2.7.4 (Uniform Roll-off)
We shall say that a sequence of functions {Fa}n*1 C Ho* uniformly rolls-off if there exists a
frequency wo clf wo(e) such that IFn(jw)| <; E for all n E Z+ for all w such that |WI > Wo.
Definition 2.7.5 (Inner Function)
Given F E '*, we say that F is inner in W* if F*(jw)F(jw) = 1 almost everywhere on the
extended imaginary axis.
RX** functions which are inner have all of their poles in the open left half plane and all of their
zeros within the open right half plane. They do not have any zeros on the extended imaginary axis.
A special class of inner functions are Blaschke products [30, pp. 63-68; pp. 132]. The following
proposition desribes their properties.
Proposition 2.7.3 (Blaschke Product)
Let {Zk}k 1 denote points in the open right half plane such that
|11-Zz|S
B(s)('
< oo. The function
-zk
k=1
is well defined and is called a Blaschke product. The Blaschke condition
Re(zk) 2-<0
k=1
1 +
zk42
is necessary and sufficient for pointwise convergence. Suppose that this condition is satisfied. B
then possesses the following properties.
(1) B is inner in H1** and |B(s)| < 1 in the open right half plane.
(2) The partial products converge uniformly on compact subsets of the open right half plane.
(3) Let K denote the compact set consisting of the points -zk- and the accumulation points of zk.
The convergence of the partial products is uniform on any closed subset of the complex plane
which is disjoint from K. B is thus analytic off K.
(4) B has an essential singularity at each point of accumulation of the zk. Hence B cannot be
extended continuously from the open right half plane to any such accumulation point, for the
extended value of B would have to be zero, while the non-tangential limits of B are of modulus 1
almost everywhere.
I
The above proposition implies that infinite Blaschke products are not continuous everywhere on
the extended imaginary axis. Hence, by proposition 2.10.1, they cannot be uniformly approximated
by Rh** functions. Moreover, B E C, if and only if B is a finite product.
Blaschke products are the only inner functions which possess zeros. We now define the notion
of a singular inner function.
Definition 2.7.6 (Singular Inner Function)
A singular innerfunction is an inner function which has no zeros.
The function F(s) = e-' is inner and it has no zeros. It is a singular inner function.
Proposition 2.7.4 (Properties of Singular Inner Functions)
Let S denote a singular inner function. Then S is uniquely determined by singular positive measure
p on the imaginary axis. S is analytic everywhere in the complex plane except at those points on
the imaginary axis which lie in the closed support of the measure pt. The function S cannot be
continuously extended from the open right half plane to any point in the closed support of p.
U
The above proposition implies that singular inner functions are not continuous everywhere on
the extended imaginary axis. Hence, by proposition 2.10.1, they cannot be uniformly approximated
by RX** functions. An example of such a function is F(s) = e-'.
Proposition 2.7.5 (Factorization of Inner Functions)
Every inner function F E hi* can be written as the product of a Blaschke product and a singular
inner function.
Definition 2.7.7 (Outer Function)
Given F E H**, we say that F is outer in R** if FXt2 is dense in 7X2 with respect to the topology
induced by the norm ||-||N on H 2 ; i.e given y E R 2 there exists x E 2 such that ||Fz - Y||i2 < E.
I
The above condition should be thought of, loosely speaking, as a "surjectivity" condition. RR1*
functions which are outer have all of their poles in the open left half plane and all of their zeros outside of the open right half plane. Such functions, in general, have zeros on the extended imaginary
axis.
Proposition 2.7.6 (Inner-Outer Factorization)
Given F E N** there exists inner and outer functions F, F, E H00 such that
F =FiF,.
This proposition can be found in [49, pp. 344]
,
[54].
Definition 2.7.8 (Minimum Phase Function)
Given F E ?*', we say that F is minimum phase if F has all of its zeros in the open left half plane.
U
2.8
Theory of Linear Operators
Let X, and 2 denote normed linear spaces over fields Y1 and F 2 , with norms |-||g1, and ||-1| A2
respectively. Let T : X1 -+ A2 denote a linear operator.
Definition 2.8.1 (Bounded Linear Operator)
We say that T is a bounded linear operatorfrom A 1 to A(2 if its induced norm, given by
||T||
=
sup
IlollAfg=1
|[Txh||
2
is finite.
The set of all bounded linear operators from K 1 to K 2 is denoted B(Af 1,A 2 ).
Definition 2.8.2 (Finite Rank Operator)
We say that T is a finite rank operatorfrom .N1 to K 2 if the closure of its range space T(K 1 ), is
finite dimensional.
The set of all finite rank operators from K 1 to K 2 is denoted B3(K 1 ,K 2 ).
Definition 2.8.3 (Compact Operator)
We say that T is a compact operatorfrom K1 to K 2 if for any bounded subset M C A1, the closure
of T(M) is compact with respect to the topology on K 2 induced by the norm||-||H2 The set of all compact operators from K1 to K 2 is denoted 3,( 1,K 2 ).
Finite rank operators are compact. Compact operators, however, are not necessarily finite rank.
Their relation to finite rank operators is given by the following proposition.
Proposition 2.8.1 (Finite Rank Approximants)
If K is either C2 or C* then Bo3(,K) is dense in B3(,K) with respect to the induced-norm
topology.
This proposition says that compact operators can usually be approximated by finite rank operators.
This proposition appears in [5, pp. 179].
2.9
Hankel/Toeplitz Operator Theory
In this section we assume that F E Lo is given.
Definition 2.9.1 (Hankel Operator)
The Hankel operator with symbol, or induced by, F is the map FF : I2 _, 12± defined by
rFg = H 2 ±Fg, where Hu21 denotes the projection operator from £2 onto N21 (or E 2(R) ).
The operatornorm of TF is given by
||TFI|
Idef
=
sup
|JFgj2-
11911L2=1
Proposition 2.9.1 (Finite Rank Hankel Operator)
F induces a finite rank Hankel operator if and only if F E "h(C+)
+ R7"(C-). The rank of such
an operator is equal to the number of open right half plane poles of F.
This proposition is known as Kronecker's Theorem and can be found in [44, pp. 46].
Proposition 2.9.2 (Compact Hankel Operator)
F induces a compact Hankel operator if and only if F E R"0(C+) + Ce(jR).
This proposition is known as Hartman's Theorem and can be found in [44, pp. 46]. The following
example illustrates the use of the theorem.
Example 2.9.1 (Application of Hartman's Theorem)
Suppose a,3, A E R are given. The Hankel operator induced by
1
F(s) =
1
as +1
is compact. The Hankel operator induced by
s +1
+
e
is non-compact.
Definition 2.9.2 (Toeplitz Operator)
The Toeplitz operator with symbol, or induced by, F is the map OF
2
2 defined by
=f Hu2Fg, where Hll2 denotes the projection operator from £2 onto X 2 (or E2 (R+) ). The
operator norm of OF is given by
||OF||
=
sup
|19114C2=
||OFg||r2-
2.10
'H' Approximation Theory
In this section we present results from R1 approximation theory. These results shall be used
throughout the thesis. We begin by presenting various notions of convergence in N**.
Let {Gn}" denote a sequence of R** functions. Also, let G be an element of *'.
Definition 2.10.1 (Uniform Convergence)
We shall say that Gn converges uniformly in R" to G, or that Gn uniformly approximates G in
R**, if
lim |IGn - G||u. = 0.
This defines the notion of uniform convergence in R**.
Definition 2.10.2 (Compact Convergence)
We shall say that the sequence {Gn} L 1 converges uniformly on all compact frequency intervals to
G, if
lim
(Gn - G)X[_o,o
= 0
Iloo
n-*o
for each 0 E R+, however large. This type of convergence shall be referred to as compact convergence
in H**.
In controlling infinite dimensional systems, an often encountered X** function is the "delay":
G(s) = e-'1A where A
> 0. The following example provides us with one method of generating
compact approximants for delays.
Example 2.10.1 (Compact Approximant for a Delay)
Let
Gn =
sA
+ n
define a sequence of RX** functions. It can be shown that the sequence defined by Gn uniformly
approximates the V** function G = e-a on compact frequency intervals [2, pp. 178].
I
The following example shows how Pade' approximants can be used to generate compact approximants for delays.
Example 2.10.2 (Pade' Approximant)
The [n, n] Pade' approximants for P(s) = e-'A are given by
Pn(s) = N,,(s)
D, (s)
where
D~(S)=n
(2n - k)! n!
D 2n!k! (n k=0
()
and
Np.(s) = D,.(-s).
We note that the P, are stable for all n. We also note that they are inner, and hence uniformly
bounded. Moreover, they uniformly approximate P on compact frequency intervals. The latter
follows from [26 where it is shown that
-
P(j)|
<;
{2(w"(e
2
|wA < 2(f)in;
elsewhere.
))2n+1
U
The following example provides us with still another method of generating compact approximants for delays.
Example 2.10.3 (Compact Approximant for a Delay)
If P(s) = e-'A, then the sequence defined by
Pn
2n - sA
2n
n
+s A)
can be shown to uniformly approximate P on compact frequency intervals. This follows from
analysis of the expression
A - 2ntan-1(-A)
|Pn(jw) - P(jw)| = 21 sin(
2n
2n2
and the fact that lim,_., 2n tan1(Q) = wtX.
In the sequel, we shall quite often deal with uniform convergence on different portions of the
imaginary axis. To deal with this type of convergence, we will need the notion of a 6-neighborhood.
Definition 2.10.3 (6-Neighborhood)
Let wo denote any finite real number. Think of wo as representing a frequency point on the imaginary
axis. By a 6-neighborhood of wo, denoted B(wo, 6), we shall mean the collection of points in the
following bounded open frequency interval
B(wo, 6)
(-wo - 6, -wo + 6) U (wo - 6, wo + 6)
where 6 > 0, however small.
The symbol B(oo, 6) will denote a 6-neighborhood of oo. It is defined as the collection of points
in the following unbounded open frequency interval
def.
B (oo, 6) = (-o 0
By symmetry B(- oo, 6) is the same set.
-
U (,
oo0).
I
With the above convention, a smaller 6 implies a "smaller neighborhood"(i.e. less points). A larger
6 implies a "larger neighborhood"(i.e. more points). Given this, it is reasonable to refer to 6 as the
size of the neighborhood B(., 6) where (.) is any positive extended real number.
We now define other notions of convergence in R**.
Let {wk} 1 denote an increasing sequence of distinct points on the extended non-negative real
axis. Here, 1 E Z+ U {oo}; i.e. the sequence can be finite or countably infinite.
Definition 2.10.4 (Convergence on Closed Sets)
We shall say that the sequence {Gn}*_ converges uniformly to G everywhere except on open
neighborhoods of the points {WO}
lim
n-* 00
, if
(Gn - G)XR/uIk=1~B(w
6)
?j00
=
0
for each 6 > 0, however small. Since the set R/ U I B(Wk, 6) is always closed, one can equivalently
say that the sequence {Gn}** converges uniformly to G on all closed frequency intervals excluding
the points {Wkly..
U
We note that the set R/ UL 1 B(o&, 6) need not be bounded; e.g. the set R/B(0, 6) = (-oo, 6] U
[6, oo) is not bounded. This unboundedness in R/ U I B(wk, 6) occurs if and only if oo is not
amongst the points {wk 1. When oo is amongst these points, the set R/ UI 1 B(wk,6) will be
bounded and hence compact. This motivates the following extension of compact convergence in
0o.
Definition 2.10.5 (Convergence on Compact Sets)
We shall say that the sequence {Gn}* converges uniformly to G on all compact frequency intervals
excluding the points {wk}5k 1 (and necessarily ±oo), if
lim1 (Gn - G)XR/B(**,6)Uu
B(wA,6) H
= 0
for each 6 > 0, however small.
Example 2.10.4 (Convergence of Sequences)
(a) The sequence defined by the function fm =
frequency intervals excluding the point 0; i.e.
lim
M-_+00
1(1 -
fm)XR/B(0,)
'H
700
=
lim
M__40
converges uniformly to unity on all closed
(1 -
fm)X(-0 0 ,-blU[ 60, 0 )
''
.OO
= 0
for each 6 > 0, however small.
(b) The sequence defined by the function gn = n converges uniformly to unity on all compact
frequency intervals; i.e.
lim1 (1 - gn)X[_,]
= 0
for each f E R+, however large.
U
In later sections we will deal with convergence in R-oo. Often we will need to turn uniform
convergence in R" on compact sets into uniform convergence. The following lemma tells us how
this can be done.
Lemma 2.10.1 (Convergence in R**)
Let F E ?** nCe. Also, let F(jWk) = 0 for each k = 1,..., 1. Suppose that {Gn}* 1 is a uniformly
bounded sequence of elements in H* . Also suppose that G E V**. If the sequence {Gn}*
1
converges uniformly to G everywhere except on open neighborhoods of the points {wk}f'1; i.e. if
lim
(Gn - G)XRut
=0
IB(wk,)
for each 6 > 0, however small, then
lim ||(Gn - G)FII
n-.+oo
= 0.
If instead, we have
=0
lim I (Gn - G)Xi..n,n]
for each f E R+, however large, then
lim
n-.+oo
II(Gn - G)FII
= 0
for each F E XYm.
Proof
The proof of this lemma follows intuitively using arguments from elementary analysis.
We shall prove the first statement for 1 = oo. The case where I is finite will then follow. The proof
of the second result will be similar.
(A) Proof of first result.
To prove the first result we will consider the quantity |(Gn - G)F| on two sets. We shall denote
the sets S6 and R/S 6 . The proof will proceed in three steps. (1) First we show that there exists a
set Sb on which F is small. We then make the quantity I(Gn - G)F small on Sb by exploiting the
boundedness of Gn and G. (2) We then exploit the boundedness of F on R/Sb to make |(Gn - G)F
small on R/S 6 by taking n sufficiently large. (3) Finally, we combine the results of (1) and (2).
Step 1: Analysis on Sb.
Let n E Z+ be fixed. Since Gn and G are uniformly bounded, there exists L E R+ such that
|(Gn - G)F < L |F
almost everywhere on the extended imaginary axis. This inequality shall be used twice below.
Since 1 = oo and the sequence {w
CA1Re is monotone, it possess an accumulation point, say
A E Re; i.e. A can be a finite non-negative value or oo 3.
3 If I = 00 then it should be noted that the domain of analyticity of F cannot. be extended to include the point
at which the wk accumulate. This follows from proposition 2.3.5. A non-constant analytic function with such
accuuiflation of zeros would necessarily be zero throughout its domain.
Since F is continuous everywhere on the extended imaginary axis, it is continuous at A. Since
F(jWk) = 0, this implies that
F(jA)
F( lim jwk) = lim F(jwk)
=
By continuity at A, there exists 6 A
= 6 A(E,
=
k-+o
k-+oo
0.
L) > 0 such that
||FXB(A,-A)
<
E
.
Combining this with the previous inequality, then gives the following result
(G, - G)FXB(A,A)
<E.-
Since A is an accumulation point of the sequence {wk}
, the set B(A, 6 A) contains all but a
finite number of the Wk. Suppose the points {wk}i 1 lie outside B(A, 6 A).
Since F is continuous at each wk and since F(jWk) = 0, then for each k = 1, 2, ... ,li, there
def
exists 6 k _ 6 k(E, wk, L) > 0 such that
FXukB(wk,bk)
From this, and the first inequality |G, - GI
L Fl, we obtain the following result
11
(Gn - G)FXUk=..B(wk,bk)
We now define
6 t
~
defAr~l1~def
6(E,A, LI ,{w1k
1) =
k=
NOO
min{ 6 A,
<
min
k=1,...li
6
k}.
Note that this minimum is well defined and strictly positive since we are minimizing over a finite
number of objects. Given this, we can define the set S6 as follows
S6 - B(A, 6) U U11 B(_k, 5).
With this definition, combining the above results gives the following
||I(G - G)FXs I1,H.
< E.
Moreover, this holds for all n E Z+. This concludes the analysis on S6 . We now turn to R/Sb.
Step 2: Analysis on R/Sb.
Since F is bounded, there exists M E R+ such that
|(Gn - G)F| :5 M |Gn - G|
almost everywhere on the extended imaginary axis. Since G,, converges to G uniformly everywhere
except on open neighborhoods of the points {wk}5 1 , we have
lim
(Gn - G)XI6
H,,
0-
This implies that there exists N
f N(E, 6, M) such that
(Gn - G)FXR/S,
,N
<
for all n > N. This completes the analysis on R/S 6 .
Step 3: Combine steps 1 and 2.
Combining the results of steps 1 and 2 gives
II(Gn - G)F||I,
for all n > N.
This proves the first result for 1 = oo. When I is finite the sequence {wk}'=. has no accumulation point and so the proof is simplified.
(B) Proof of second result.
The proof of the second result is similar to that of the first. Since Gn and G are uniformly
bounded we have I(Gn - G)FI < L |F| almost everywhere on the extended imaginary axis. Since
F E '-O, there exists fl 4f (c, L) > 0 such that
(Gn
-
G)FXB(O,
<E
K-
for any n E Z+. By assumption, lina-., (Gn - G)Xn,oO
= limn-_, I(Gn
- G)XR/B(oo,A)
0. Since F is bounded we have |(Gn - G)FI 5 M |Gn - Gj almost everywhere on the extended
imaginary axis. This then implies that
lim
,
(Gn - G)FX
= 0.
Combining the above gives
lim ||(Gn - G)FI.
= 0.
U
This proves the second result and completes the proof.
This lemma will be applied often in sections to come. The key to the lemma is that F, loosely
speaking, "rolls-off" on portions of the imaginary axis where the convergence of Gn to G may be
poor. In light of this, we see that roll-off is desirable in certain situations. We thus define the
following special roll-off functions.
Definition 2.10.6 (Roll-off Functions)
def
fm ()
gn(s)
-,
Y2
(S2
def
q
fs
a
n
s=
s +aa
+2k±+b2V
+2bs+w
/
)T =1S2
+ 2I+2+12
)1
2+W
12
Nmk
q
w a R ,qa
2q
where a, b E R+, 11, 12, Mk E Z+ are fixed, and m, n, q E Z+ .
+ q
I
These functions shall be instrumental in the work that follows. It is important to note several
properties about the functions fm, gn, and hq.
First note that they each belong to R** and have magnitudes, and hence norms, which are less
than or equal to unity. Moreover, they are continuous everywhere on the extended imaginary axis.
Secondly, one should note that they each approximate unity as follows.
The sequence {fm'=1 uniformly converges to unity on all compact frequency intervals; i.e.
lim
=0
(1fm)X[-ofl]
for each 0 E R+, however large. The sequence {g,}** uniformly converges to unity everywhere
except on open neighborhoods of the single point Wk; i.e.
lim
(1 - gn)XR/B(wk,5)
0
=
for each 6 > 0, however small. The sequence {h}**_ uniformly converges to unity on all compact
frequency intervals excluding the points {wk}k21 (and oo); i.e.
hX=0
lim (1 - h,)X/
q-.+ORBO')U1
)UUtB(w&,6)
~kb
for each 6 > 0, however small.
The irrational functions fm and ga have appeared in [14]-[16], [53]. They possess the special
property that their phase can be made arbitrarily small, uniformly in frequency, by taking m and
n, respectively, sufficiently large; i.e.
mlim
||6/m||cjo
= 0
and
lim ||6,.||c
=
0.
We note, for example, that
r
-arctan
|im
=
M=jw]
<
2m
This special phase property of f m and gn shall be very important in what follows.
The rational function hq has appeared in [56], [59],[60]. It does not possess the special phase
properties that fm and gn possess.
The following lenuna is analogous to lemma 2.10.1. It will allow us to turn compact convergence
in R** into uniform convergence in - 2.
Lemma 2.10.2 (Convergence in H 2 )
Suppose that Gn, G E RO*, where the Gn are uniformly bounded in Hr*. If
lim 1(Gn - G)XO,0
for each 0 E R+, then
lim
for each F E R2,
II(Gn - G)F| 17 2 =
=
0
0
I
2
Proof The proof of this follows using the natural roll-off of H functions and arguments similar
to those used in the proof of lenuna 2.10.1. We have
II(Gn - G)F||
= -
< (G
-
G)X[-o,n]
2
=
I(Gf - G)F| 2 dw
2 0
|(
-
G)F|2 dw
-
|(Gf - G)F| 2dw
+ -
IF|2dw + |IGn
-
GI
IF|
2 dw.
Since Gn and G are uniformly bounded in H*, there exists M E R+ such that
I{Gn - G)F| 122
IF 2dw.
(Gn - G)X[_n,n] 2|F|12 + M
Since F C H 2 , the second term can be made arbitrarily small by taking 0 sufficiently large; i.e.
M, F) E R+ such that
there exists f0 , t e(e,
II(Gn - G)FI| 2 2
(Gn - G)X[_n.,n]
AIF|122
+
e
for all n E Z+. Given this, and the assumption, the first term can be made arbitrarily small by
taking n sufficiently large; i.e. there exists N ' N(E, fl,, F) E Z+ such that
II(Gn - G)FI||
2
2E
for all n > N. This proves the result.
In the sequel we will require the following proposition on the uniform approximation of H"
functions by R?** functions. It tells us, for example, that the irrational function f, can be
approximated uniformly in N** by RX** functions.
Proposition 2.10.1 (Existence of RX* Approximants)
A function F E X** can be uniformly approximated by RX** functions if and only if F E Ce.
Suppose F E C.. In such a case, the approximants can be constructed such that they roll-off if and
only if F rolls-off.
If in addition, F is minimum phase, then the approximants can be constructed such that they
are minimum phase4 .
The approximants can be constructed such that they are inner if and only if F is inner.
Finally, suppose F is also outer with a finite number of zeros on the extended imaginary axis;
each with finite algebraic multiplicity. In such a case the approximants can be constructed such
that they are minimum phase with the exception of zeros of multiplicity 1 on the imaginary axis
at the imaginary zeros of F.
U
Proof
The key elements of this proposition can be found in [28] and [30, pp. 17 - 183.
Related to this proposition are the theorems of Runge, Mittag-Leffler, and Mergelyan [49].
I
The following example illustrates how one might approximate an outer function with minimum
phase functions.
'Minimui phase functions have all of their zeros in the open left half plane.
Example 2.10.5 (Approximation of an Outer Function)
-.
Let F(s) = V 1 We note that F E H,** n C, and is outer. We now show how F can be
uniformly approximated by RR' functions which are minimum phase with the exception of a zero
at a = 0 with multiplicity 1.
The sequence defined by F, f G,, where G, d-f
TQ,7;1; 7"gT approximates F uniformly in
?**. The function G, is minimum phase, and can be approximated uniformly by minimum phase
RX** functions 0,5. Given this, the sequence defined by P, tef a d0 also uniformly approximates
F.
The following proposition can be found in [2, pp. 176-177].
Proposition 2.10.2 (Compact Convergence and Analyticity)
Let V be a domain in the complex plane. Let {F},j
be a sequence of analytic functions on D
which converge uniformly to F on all compact subsets of D. It then follows that F is analytic on
D. Moreover, the derivatives of Fn (of any order) converge to those of F uniformly on all compact
subsets of D.
I
2.11
Duality Theory
Let A be a normed linear space, over a field F, with norm |1-|1 . Duality theory deals with the
study of bounded linear functionals.
Definition 2.11.1 (Bounded Linear Functional)
Let f : A -+ F be a linear map. Such a map is called a linearfunctional on
bounded linearfunctional on K, if its induced norm
def
|||I'il
sup
K.
We say that
f
is a
If(X)|
ll4ll=1
is finite.
Definition 2.11.2 (Dual Space)
The set of all bounded linear functionals on K is denoted K*. This set defines a normed linear
space over the field F, when endowed with the induced-norm. It is called the dual space of K. The
space K is often called the primal space.
Definition 2.11.3 (Weak* Convergence)
Let {fn,}**n1, f C K*. We say that fn converges to
f in the weak* topology on K*, if
lim |fn(g) - f (g)|= 0
for all g E K. In such a case we say that
f
is the weak* limit of
fn.
The above shows that weak* convergence is nothing more than pointwise convergence on the primal
space K.
Proposition 2.11.1 (Implication of Weak* Convergence)
Suppose {fn}* 1 , f C K* and that fn is weak* convergent to f. It then follows that
||f||g. ; n-+oo
lm k>nf ||fkIMg..
A proof of this can be found in [5]. It is based on the Uniform Boundedness (Banach-Steinhauss)
Theorem. This "deep" result is extremely powerful. It shall be exploited in subsequent chapters.
For normed linear spaces the following proposition provides a complete characterization of
compactness.
Proposition 2.11.2 (Compact Set)
A subset of K is compact if and only if every sequence in K has a subsequence which converges to
a point in K. In such a case one says the the subset possesses the Bolzano- Weierstrass property.
U
The above is not true for general topological spaces. Some authors use the phrase sequential
compactness to characterize the above behavior. Since we shall only be working with normed linear
spaces, such preciseness is not required.
Whether a normed linear space is finite dimensional or infinite dimensional can be determined
by ascertaining whether or not its closed unit ball is compact. This is seen from the following
proposition.
Proposition 2.11.3 (Finite Dimensionality)
K is finite dimensional if and only if its closed unit ball is compact.
This proposition shows that infinite dimensional spaces necessarily have non-compact closed unit
balls.
The following proposition gives us a very useful topological property of the closed unit ball in
the dual space *.
Proposition 2.11.4 (Alaoglu's Theorem)
The closed unit ball in K* is weak* compact; i.e every subsequence has a weak* convergent subsequence.
This proposition can often be used to prove the existence of certain limits. Its proof can be found
in [5, pp. 134].
The following proposition is an application of Alaoglu's theorem.
Proposition 2.11.5 (On Weak* Convergence)
Suppose S C I* is weak* compact and {fn},*=1 c S. Suppose also that every weak* convergent
subsequence of {fn}, 1 converges to L E S. Then, fn has L as its weak* limit.
Proof
The proof of this proposition follows readily using a contradiction argument.
Suppose that L is not the weak* limit of fn. If this is the case then there exists a subsequence
{fn(k)}
1
and an element g E NV such that
Ifn(k)(g) - L(g)| I
e
for all k.
The subsequence {fn(k)}"'1 is contained in the compact set S. It is thus uniformly bounded
in A*. By Alaoglu's theorem it follows that it possesses a weak* convergent subsequence. By
assumption this subsequence must converge to L. This contradicts the above inequality. It must
therefore be that L is, in fact, the weak* limit of the sequence {fn},'* 1 . This completes the proof. I
An analogous proposition can be proved for other topologies and notions of convergence. The
idea is intuitive: given compactness and a unique limit point guarantees convergence to the limit
point.
2.12
Equicontinuity, Normality, Arzela-Ascoli
In this section we shall follow the developement given in [49, pp. 245, 281-282]. Let F denote a
family of complex-valued functions defined on a domain D in the complex plane.
Definition 2.12.1 (Equicontinuous Family)
We say that F is equicontinuouson D, if given E > 0, there exists 6 4 6(e) > 0, such that
sup If(sl) - f(s2)| < e
for all S1, S2 E D satisfying
Isi
- S21 < 6-
We note that the functions of an equicontinuous family are necessarily uniformly continuous.
Definition 2.12.2 (Pointwise Bounded Family)
We say that F is pointwise bounded on D, if given s E D, there exist M(s) < oo, such that
sup If(s)|
M(s).
Definition 2.12.3 (Uniformly Bounded Family on Compact Sets)
We say that F is uniformly bounded on compact subsets of D, if given a compact subset S C D,
there exists M(S) < oo, such that
sup
{fED;
SES}
If(s)I
M(s).
Definition 2.12.4 (Normal Family)
We say that F is normal on D, if every sequence of functions in F contains a subsequence which
converges uniformly on all compact subsets of D. In such a case we say that every sequence in F
exhibits the Bolzano-Weierstrauss property on compact subsets of D.
The following specialized version of the A rzela-Ascoli theorem gives sufficient conditions for
normality.
Proposition 2.12.1 (Arzela-Ascoli Theorem)
Let F be a family of complex-valued analytic functions defined on a domain D in the complex
plane. If F is uniformly bounded on each compact subset of D, then F is normal on D.
I
The following example illustrates how the Arzela-Ascoli Theorem may be applied. Arguments
similar to those used in the example shall play a critical role in the later chapters.
Example 2.12.1 (Application)
Let {Z,}*,I C IX* be a uniformly bounded sequence. The above proposition implies that {Zn}**_
is a normal family. Consequently, there exists a subsequence {Zn(k)}'l which converges uniformly,
to say L, on compact subsets in the open right half plane. Since Zn(k) is uniformly bounded in
XH*, L will be bounded in the open right half plane. From proposition 2.10.2, it then follows that
L Eh *.
2.13
Summary
In this chapter we established notation to be used throughout the thesis. The spaces H** and H2
were defined and discussed. Results from ?** approximation theory were also presented.
Chapter 3
Results from Algebraic Systems
Theory
3.1
Introduction
In this chapter we present several results from algebraic systems theory.
3.2
The Youla et al. Parameterization
Throughout the thesis, we shall focus on feedback systems having the classical structure shown in
Figure 3.1.
Here F denotes a plant and K denotes a compensator. We shall assume that both are causal,
linear time invariant, single-input single-output, continuous time systems. At the moment, we make
no assumption about their dimensions; i.e. each can be finite dimensional or infinite dimensional
systems. r and d denote exogenous signals. e and u denote the errorand control signals, respectively.
Let H(F,K) denote the transfer function matrix from r, d to e, u. We then have
e
where
H(FK)
i
1
1--FK
,
F
1-FK
Given this, we have the following terminology. The transfer function
K shall be referred
to as the sensitivity transfer function. The transfer function -FKK shall be referred to as the
complementary sensitivity transfer function. Finally, the transfer function KK shall be referred
to as the reference to control transferfunction.
Let 1Z denote some class of scalar functions which we shall refer to as the stable transfer functions
(e.g. 7H** ). Let M(R) denote the set of all transfer function matrices with elements in R.
We shall say that the feedback system in Figure 3.1 is R-internally stable or that K internally
stabilizes F with respect to 1? if H(F,K) GM(1R).
SR(F) will be used to denote the set of all compensators K which internally stabilize the plant
F with respect to R. The class 1? should be viewed as a ring. We now define this and other
algebraic concepts [56].
Figure 3.1: Feedback Control System Structure
Definition 3.2.1 (Algebraic Concepts)
A ring is a collection of elements 1? on which we define two binary operations + : 7? x R -+ Z and
- : R x R -- R. Given any x, y, z E R, the operations
+
and - satisfy the following ring axioms.
(1) Closure: x + y E 7?.
(2) Comnutivity: x + y = y + z.
(3) Associativity: (z + y) + z = x + (y + z).
(4) There exists an element 0 E 1? called the additive identity, such that z + 0 = x.
(5) There exists an element -x E R called the additive inverse of z, such that x + (-X) = 0.
(6) Closure: z - y E 1?.
(7) Associativity: (x - y) - : = x - (y - z).
(8) There exists an element 1 E R called the multiplicative identity, such that x - 1 = 1 - X = x.
The existence of such a multiplicative identity is what algebraists mean when they say that a ring
7? contains an identity.
(9) Distributivity: x - (y + z) = x . y + x - Z.
(10) Distributivity: (z + y) - z = x . z + y - z.
We say that R is a commutative ring if the binary operation (-) is commutative on R; i.e.
z - y = y . z.
Let 7? be a commutative ring. We say that 7? is a domain if x, y # 0 implies that z - y # 0 and
y -x # 0. x is a unit of y if x -y = 1 and y - x = 1. In such a case, we say that x and y are invertible
in R.
Given this, we have the following parameterization for Sn(F) [56, pp.364], [58], [591. We refer
to it as the Youila et al. parameterization.
El
Proposition 3.2.1 (Youla et al. Parameterization)
Let 1Z be a commutative domain with an identity. Let F(1R) denote the fraction field associated
with R. Suppose F E F(1R). Suppose also that F has a coprime factorization (N 1 , D1 ) over the
ring R with Bezout factors (Nk, Dk); i.e. suppose there exists elements N1 , D1 , Nk, Dk E 1Z such
that
N
D1
and
(D 1 Dk - NfN)-' E 1Z
for all s in the domain of definition of R. Given this, the set of all compensators K which internally
stabilize F with respect to 1Z can be parameterized as follows:
SR(F) =
Nk- DQ
Dk
- N1
Q
IQ
E I.
Throughout the thesis, the symbol P shall denote an infinite dimensional plant 2 to be controlled.
It shall always be assumed that P satisfies the assumptions of proposition 3.2.1. This shall be stated
formally in assumption 4.2.1.
Given this, we shall often write K(P, Q) to denote a particular compensator which stabilizes P.
With this notatation, K(P, -) represents a bijection from R to Sg(P).
The utility of the above parameterization is that it allows us to express the individual transfer
functions of H(P,K(P,Q)) as affine functions in the parameter Q:
H (P, K(P,Q)) =
Dp(Dk - NpQ)
Dp(Nk - DpQ)
Np(Dk - NQ)
Dp(Dk - NQ)
This affine dependence on Q is invaluable in optimization problems.
Finally, we note that if P E "R,then we can choose N, = P, Dp = 1 ,Nk = 0 , and Dk= 1.
The set of compensators K which internally stabilize P can then be parameterized as follows
-Q
K(P,Q)= 1-PQI
def
where Q can be any element in R.
3.3
A Stability Result
In what follows, we shall require the following proposition on internal stability.
Proposition 3.3.1 (Internal Stability)
Let R denote a commutative domain with an identity. Let F denote a plant and C a compensator.
Let (N, Df) E R be coprime factors for F over R. Let (Nc, De) C R be coprime factors for C
over R. We then have that C internally stabilizes F with respect to the ring 1? if and only if
6(F,C) =' DiDe - NNe isa unit of R.
'It should be noted that the compensators in Sx(F) need not be causal [17, pp. 3]. [?). Conditions will be given
so that this is avoided. All compensators which we construct in the thesis, of course, will be causal.
2
By infinite dimensional we mean not necessarily finite dimensional.
El
N
Proof The proof in [56, pp. 45-46, lemma 4] applies to our general ring 1?. The main ideas are
now given.
If 6 is a unit in 7?, then the associated closed loop transfer function matrix
H(F,C)=
NS Dc
1
D Ne D1 De
6
D De
has all of its transfer functions in R; i.e. H(F,C) E M(R). This proves one direction.
If H(F,C) E M(R), then we have
[
f
Dc Nc ]E M(R).
The coprimeness of (NS, Df) and (Ne, Dc) then implies that
and hence completes the proof.
3.4
j E 7?.
This proves the other direction
The Corona Theorem
We begin this section with an instructive example.
as a Ring)
Example 3.4.1 (?1X
The function space X"t is a commutative domain with an identity [56]. It shall be used throughout
the thesis to define stability; i.e. it will be the the only ring over which we will work.
Moreover, we shall always assume that our plant P belongs to the fraction field associated with
X"O. We shall also assume that it has a coprime factorization over the ring 3. With this assumption,
we will be able to use the parameterization in proposition 3.2.1. This, in turn, will allow us to
express system transfer functions as affine functions with respect to the parameter Q.
When we deal with the ring H we are implicitly concerned with 2 - finite gain stability as
defined in [11].
Given this, the following proposition will allow us to determine whether or not two functions
in X" are coprime.
Proposition 3.4.1 (Corona Theorem)
Given N, D E 7? where R is X", then N and D are coprime in R if and only if
inf IN(s)| + ID(s)I > 0.
3EC+
This proposition can be found in [56, pp. 341-342]. It is sometimes referred to as the Corona
Theorem. Loosely speaking, it says that N and D are coprime in 1? if and only if they have no
common open right half plane zeros.
"It should be noted that not all functions in the fraction field of A have coprime factorizations [56]. In contrast,
all stabilizable functions in the fraction field of 7* have coprime factorizations [52]. This shall receive further
consideration in the sequel.
3.5
Summary
In this chapter we presented various results from the algebraic systems literature. We are now
ready to precisely state what fundamental problems shall be addressed in this thesis.
Chapter 4
Statement of Fundamental Problems
4.1
Introduction
In this chapter we precisely define what we shall refer to as the K-Norm Approximate/Design
J-Problem. This problem shall be the focal point of the thesis. This problem addresses the questions: What is a "good" plant approximant? How do we obtain a near-optimal finite dimensional
compensator?
We also define two additional problems, the K-Norm Purely Finite DimensionalJ-Problemand
the K-Norm Loop Convergence J-Problem. The first problem addresses the computation of optimal
performance measures using finite dimensional techniques. The latter, examines the convergence
properties of designs which result from the Approzimate/Design approach taken in this thesis.
4.2
Basic Assumptions & Notation
In this section we shall make assumptions which will allow us to use the algebraic system-theoretic
results of Chapter 3. In doing so we shall establish notation to be used throughout the thesis.
Let P(s) denote the transfer function of a continuous time scalar infinite dimensional plant 1.
Also, let {Pn(s)} 1 denote a sequence of finite dimensional (real-rational) approximants of P. The
sense in which Pn approximates P shall be stated in the sequel.
Throughout the thesis we shall assume that the plant P and the approximants Pn satisfy the
assumptions in proposition 3.2.1. More precisely, we make the following assumption.
Assumption 4.2.1 (Permissible Plants and Approximants)
Let F(R) denote the fraction field of a commutative domain 2 R with an identity (e.g. h"o). Let
Fe(R) denote those elements of F(R) which are expressible as the ratio of coprime factors in R.
It shall be assumed that the plant P and the approximants {Pn(s)}'o are elements of Fe(R).
Given this, P has a coprime factorization over the ring R; i.e. there exists elements N,, D,, Nk, Dk E
R such that
P = -- ?
D,
'By infinite dimensional we inean not necessarily finite dimensional. Although not, explicitly stated. causality and
linear time invariance are implied.
2
A domain is a ring such that the product of non-zero elements is non-zero.
and
DDk
NpNk = 1
-
for all s in the domain of definition of R. It then follows from proposition 3.2.1 that the set of all
compensators K which internally stabilize P with respect to the ring R can be parameterized as
follows:
K(P,Q) = Nk - DPQ
Dk - NQ
where Q E R. Given this, K(P,.) is a bijection from R to the set of all compensators SIZ(P) which
internally stabilize P with respect to R.
Also, assumption 4.2.1 allows us to associate with each P, the elements N,,, D,,, Nk., Dk, E 7?
where
Pn =
N''
Dn
and
D,,Dk, -
NpNk, = 1
for all s in the domain of definition of R. Again, from proposition 3.2.1, it follows that the set of all
compensators Kn which internally stabilize Pn with respect to 7? can be parameterized as follows:
K(Pn, Q) = Nk,
-
D,,Q
Dk, - N,,Q
where
4.3
Q E R.
A/-Norm Approximate/Design J-Problem
In this section we precisely define the /-Norm Approzimate/Design J-Problem. First, we shall
need some assumptions and definitions.
Given the discussion in the previous section, we let JA((., K(.,-)): Fe(R) x Fe(7) x R -+ R+
denote a performance measure. We shall refer to it as the K-norm J-measure. This terminology
is used since in the sequel, K will denote the norm on one of the function spaces (X**, XJ2 ) and J
will be represent either a sensitivity criterion or a mized-sensitivity criterion.
Given this, the optimal performance for the infinite dimensional plant P with respect to the
K-norm J-measure shall be defined as follows.
Definition 4.3.1 (Optimal Performance)
ppt
def
infJg(P,K(P, Q)).
QEIZ
The optimal 3 solution to this problem shall be denoted Qpt. The corresponding compensator
will be denoted Kpt and is given by
def
Nk
-D~p
Kopt= K(P, Qp ) = Nk- DQopt
D
ahIi may be that an optimal solution does not exist.
This
-
NQ 0
it
technical point is nmot important at tihe moment.
As stated earlier, our primary motive is that of finding a near-optimal finite dimensional compensator for the infinite dimensional plant P. To do so, one starts by examining the optimization
problem defined in definition 4.3.1.
In general, definition 4.3.1 defines an infinite dimensional optimization problem which is very
difficult to solve. The basic philosophy of this research endeavour has been to avoid the Design/Approximate approach which appears in the literature. In this approach, one first needs to
solve the above infinite dimensional problem for an infinite dimensional compensator. Then, one
approximates the resulting infinite dimensional compensator to get a desirable near-optimal finite dimensional compensator [14]-[16], [40]-[41]. Rather than this approach, we have taken an
Approximate/Design approach.
In our Approximate/Designapproach, we first "approximate" the plant P with a sequence of finite dimensional approximants {Pn}* 1 . Then, to obtain a suitable finite dimensional compensator,
we consider the following finite dimensional optimization problem.
Definition 4.3.2 (Expected Performance)
pn te inf Jg(P, K(P., Q)).
QEIZ
In the context of this work, we shall refer to pn as the expected performance. This terminology for
pn is motivated by the fact that the numbers pin are typically used to guide us during the design
process.
Let Q, denote the optimal 4 solution for the problem in definition 5.5.1. In accordance with
assumption 4.2.1, Q, generates an internally stabilizing 5 compensator K, for P. This compensator
is given by:
N
Kn = K(P,,Q) =
D
,-
Dk, - NnQn
It is shown in section 6.3 that, in general K, need not stabilize P, even when Pn is "close" to
P in the uniform topology on 1? 6. Consequently Kn, in general, cannot be placed in a feedback
loop with the infinite dimensional plant P. It is thus appropriate to ask whether or not K, can
be modified to obtain a near-optimal finite dimensional compensator for P. We shall see that for
a large class of problems, the answer to this question is affirmative.
Toward this end, we define a "roll-off operator"
r: Q, -- Q
which maps Q, to Qn E 1?. From the above discussion, it is clear that the roll-off operator r must
be chosen intelligently (see section 6.3). The exact form of r will be determined shortly.
Given this, we then consider the feedback system in Figure 4.1, obtained by substituting the
finite dimensional compensator generated by
-
def
Qn,
-
namely
Nk,.
-
D~.Q,.
Dk,
-
NQ
Kn = K(PnQn) = N - D,,Qn
"It may be that an optimal solution does not exist. Again, this technical point is not important at the moment.
5
We note that this may not be the case since Q. may not lie in the ring R. This point. is also associated with
existence issues and is not critical at the moment.
6
Since we shall be working over the ring 7", it is clear what is met by the uniform topology.
into a closed loop system with the infinite dimensional plant P. We note that although k, internally
stabilizes P, it need not internally stabilize P. This critical issue will be addressed in the sequel.
Let H(P,k,) denote the resulting closed loop transfer function matrix from r, d to e, u. We
then have
e H(P,kn)
,
where
1-PR,
1-PR,
Substituting for k, then gives
H(P,kn(On)) =
D,(Dk - N,,Qn)
Dp(Nk, - D,,Qn)
N,(Dk, - N,,Qn)
Dp(Dk, - NQn)
6(P,kn(Qn))
where
6(P,kn(Qn)) t= DP(Dk, - N,,Qn) - NP(Nk, - D,,Qn).
Assuming internal stability can be shown, we then have the following "natural" definition for
the actual performance.
Definition 4.3.3 (Actual Performance)
-def
, = JW(P, K,)-
U
Comment 4.3.1 (Internal Stability)
AdefSince
_- Nk, - D,,QnDk, - N,,Qn internally stabilizes Pn, the factors Nk, - D,,Qn and
Dk, - N,,Qn must be coprime. This follows from proposition 3.3.1. Given this, it follows from
the same proposition, that kn(Qn) internally stabilizes P if and only if
1
E R. This
O(P,R,(O,))
argument will be used in the sequel.
Given the above discussion, we define the K-Norm Approximate/Design J-Problemas follows.
Problem 4.3.1 (Approximate/Design Problem)
Find conditions on the approximants {P},*1 and on the roll-off operatorr such that
lim ftn = popt.
n--+oo
If a set of approximants {Pn}** 1 satisfy the above condition, then we shall say that the approximants are "good"; i.e. a set of approximants {P}o_1 are good if they allow us to satisfy the
control objective. In this sense then, the K-Norm Approximate/Design J-Problem addresses the
question: What is a "good" finite dimensional approximant?
1 0 pt J (P Kopt)
/-Ln=J(Fn,Kn)
an= J(P Kn)
Figure 4.1: Visualization of Approximate/Design Methodology
Assuming k, internally stabilizes P, we have
Popt
An.
This is because popt is the optimal performance. Given this, we would like to show that
lim n <_ pti.
Doing so, however, is equivalent to finding an internally stabilizing finite dimensional compensator
Kn such that
Tin 5 PL0 t
+E
where e > 0 is a given (apriori) optimality tolerance. In this sense, the K-Norm Approximate/Design J-Problem is equivalent to that of finding a near-optimal finite dimensional compensator for the infinite dimensional plant P.
This K-Norm Approximate/Design J-Problem shall be the focal point of the thesis. More
specifically, the problem shall be solved in the sequel for two distinct performance measures Jg.
To define these measures we shall need weighting functions W, W 1 , W 2 E R.
Let K denote one of the function spaces H** and H 2 . Suppose that F, G E F,(R) and Q E R.
If the performance measure J has the form
Jg(F,K(G, Q)) def
=
W
,
1 - FK(G, Q) yV
then problem 4.3.1 will be referred to as an K-Norm Approximate/Design Sensitivity Problem. A
solution to this problem shall be presented in Chapter 6 when the norm||-||g is the X**o-norm. The
X 2 -norm case is addressed in Chapter 8.
If the performance measure Jg has the form
[W1
Jg(F,K(G, Q))
def
W2 FK(G, Q)
1-FK(GQ)
or
Jg(F,K(G, Q))
=
Q)def
iW1
WV2K(G, Q)
1 - FK(G, Q)
,
then problem 4.3.1 will be referred to as an K-Norm Approximate/Design Mixed-Sensitivity Problem. A solution to this problem shall be presented in Chapter 7 when the norm ||-||g is the
H**-norm. The X 2 -norm case is addressed in Chapter 8.
The above Approximate/Design problems are difficult for several reasons. These reasons are
discussed in section 6.3.
4.4
K-Norm Purely Finite Dimensional J-Problem
In practice we would like to be able to compute popt using finite dimensional algorithms. For the
above sensitivity/mixed-sensitivity criteria, the literature has typically considered infinite dimensional eigenvalue/eigenfunction problems in order to compute popt [14, pp. 28-31], [63, pp. 308].
With an ultimate intention of providing purely finite dimensional techniques for computing pot,
we shall also consider the following "purely" finite dimensional problem.
El
Problem 4.4.1 (Purely Finite Dimensional Problem)
Find conditions on the approximants {Pn}oo such that
lim pn = p4opt.
In the context of this work, we shall refer to this problem as the K-Norm Purely Finite Dimensional
J-Problem.
This problem inherently addresses the question of computing the optimal performance p, using
finite dimensional techniques.
Again, if Jg is a sensitivity criterion then we will refer to this problem as the K-Norm Purely
Finite Dimensional Sensitivity Problem. Analogously, if JA( is a mixed-sensitivity criterion then
we will refer to this problem as the K-Norm Purely Finite DimensionalMixed-Sensitivity Problem.
These problems shall also be addressed in Chapters 6 and 7 for w* criteria. Chapter 8 shall address
the problem for the case of an 7j2 performance criteria.
4.5
KN-Norm Loop Convergence J-Problem
Having the actualperformancefin and expected performance pn converge to the optimal performance
popt is quite desirable. However, given this it is still very important to understand in what sense
the transfer functions associated with the actual system (P, kn) converge to those of the optimal
system (P, Kt).
Given this, we define the K-Norm Loop Convergence J-Problemas follows.
Problem 4.5.1 (Loop Convergence Problem)
Given that limno An = pt and limn_.o pn = popt, in what sense, if any, do the transferfunctions
associated with the (P, kn)-loop converge to those of the optimal (P, Kopt)-loop ?
U
In this problem we concern ourselves primarily with the convergence properties of designs which
result from the Approximate/Design approach taken in this thesis.
This problem shall be addressed in the sequel for each of the afforementioned design criteria.
4.6
Summary
In this chapter three problems were defined: (1) The K-Norm Approximate/Design J-Problem,
(2) The K-Norm Purely Finite DimensionalJ-Problem, and (3) The K-Norm Loop Convergence
J-Problem.
In the sequel we shall address each of these problems for R"* and R 2 sensitivity and mixedsensitivity design criteria.
Chapter 5
HOO Model Matching Problem
5.1
Introduction
In this chapter we define the R** Model Matching Problem. It is shown how near-optimal realrational solutions can be constructed for this problem by considering sequences of appropriately
formulated finite dimensional model matching problems. The constructions presented shall be
heavily exploited in subsequent chapters on design via N* optimization.
5.2
Infinite Dimensional HiX
Model Matching Problem
In subsequent chapters, we will be examining optimization problems which have the following
structure.
Definition 5.2.1
pt
def
inf |IT, -
QENjO
T2QII
where T1, T2 E R .
Given this, the R** Model Matching Problem is that of finding a near-optimalQ-parameterin
1.o.
I
Since T2 E ?* , we know from proposition 2.7.6 that it possesses an inner-outer factorization
over X**. We will denote this factorization by T2 = T2iT2., where T2, is inner in H** and T2, is
outer in h**. This factorization shall be exploited throughout the chapter.
Throughout this chapter, the following technical assumption shall be made on T, and T2.
Assumption 5.2.1 (T1 and T2)
(1) T1 E Ce 1.
(2) T2, has a finite number of zeros on the extended imaginary axis; each with finite algebraic
multiplicity. See comment 5.2.1 below.
(3) If T2. is not invertible in R*,
1Functions
then we assume that T2. E Ce.
in Ce are continuous everywhere on the extended imaginary axis.
The following comment establishes notation regarding the imaginary zeros of T2.. It also clarifies
what we mean by each zero having finite algebraic multiplicity.
Comment 5.2.1 (Imaginary Zeros of T24: Finite Algebraic Multiplicity)
We shall denote the zeros of T 2. on the extended imaginary axis by the finite sequence {Wk} =0 . It
shall be assumed that the sequence is increasing and that its elements are distinct, non-negative,
extended real numbers. We shall also adopt the convention that WO = 0 and w1 = oo. Given this,
{wk} ~1 will denote finite zeros of T2. on the positive imaginary axis. {-wk} ~1 will denote finite
zeros of T2. on the negative imaginary axis.
By assumption 5.2.1, each Wk has finite algebraic multiplicity. This multiplicity shall be denoted
mk, is non-negative, and is generally a fraction.
Since T2, E Ce, we know that it is bounded away from zero on all closed frequency intervals not
containing an imaginary zero. Consequently, if we define
d-1
F(s) =ms"O
](s
2
+ o2)"h
k=1
and
Ga(s)
def
(
\s
a
+ a/
where a E R+, then
1
TZ'(-) F(.) F-1(. + -) Ga(-),
aa
T 2.(.) F-'(.) G
(.) E 'H*
OfCe
for each a E R+.
Several points need to be made in order to emphasize the broad applicability of assumption
5.2.1 above.
Comment 5.2.2 (Broad Applicability)
(1) T1 and T2 may be real-rational.
(2) T2j need not be continuous on the extended imaginary axis as assumed in [60]. It may, for
example, have a singular inner part; e.g. T2, = e~
(3) T2j may have an infinite Blaschke product [30, pp. 132]; e.g.
*I1-z~I
T2;
k=1
where {zk}'
s-zk
-~k s+ 'k
are open right half plane zeros satisfying the Blaschke condition
Re(zk)
=1
+
Z 2
(4) The zeros of T2, on the extended imaginary axis, strictly speaking, may be branch points
=
of T2. with respect to the entire complex plane; e.g. T2.
.
In such a case we assume a
single-valued analytic branch in the open right half plane.
(5) T2 may possess removable singularities, poles, essential singularities, and branch points in
the open left half plane.
(6) Although T2 , E Ce, this does not mean that T 2 E Ce; e.g. T2 = e - (
(7) Although the outer function T2. =
) (-).
_, is not continuous on the extended imaginary
axis, it is admissible since it is invertible in R**; i.e. 1 - e-' E R**.
(8) Cases in which the associated Hankel operator rTrT2. is non-compact are permissible; e.g.
Ti= N, T2i = e~' (cf. proposition 2.9.2).
The above Model Matching Problem should be viewed as an inversion problem. This is because
in posing the problem we indirectly tell the mathematics to find an admissible Q which inverts as
much of T 2 as possible. The problem should thus be approached with an "inversion mentality".
In this section we shall construct a sequence {Q,} m =1 C ?o* such that given an optimality
tolerance e > 0, however small,
Po
|IT 1 - T 2Qm||u.i
Yo +
C
for m sufficiently large. This will allow us to obtain a near-optimal Q-parameter and thus provide
a solution to the NX** Model Matching Problem. To construct this sequence, we first shed light on
exactly what part of T2 should be inverted.
From the above inner-outer factorization for T 2 , we immediately note that, in general, T2i is
not invertible in R**. We also note that any zeros which T 2, may have on the imaginary axis, are
also non-invertible. This leads us to conjecture that we should invert T2, away from its imaginary
zeros in the closed right half plane. This statement/conjecture shall be formalized and confirmed
shortly.
In order to gain insight into the 0i* Model Matching Problem, one usually starts with some
version of the following proposition [23], [44, pp. 46]. The proposition emphasizes (and quantifies)
the non-invertibility of T2i.
Proposition 5.2.1 (Inner Problem)
Each of the following are equal:
inf ||JT1 - T2i Z||h.(1
ZEN**
(2)
Moreover, there exists Z, E
7VO,
rTrT
not necessarily unique, which attains the infinum in the "inner
problem" defined by (1); i.e.
|IT1 - T2iZo| . = mn IT1 - T2iZ||o.
In the sequel, we shall refer to Zo as the "inner solution".
Proof The existence of an infimizer Z0 can be proved using standard weak* convergence ar uments [5, pp. 134-137), [35, pp. 128), [24, pp. 85 - 86). The fact that the infimum is FTTi; is
essentially a special case of Nehari's Theorem [38]. Elements of this proposition can also be found
in [1], [39], [46], [51].
U
Typically, one solves the "inner problem", defined in proposition 5.2.1, in order to get a handle
on the the optimization problem defined in definition 5.2.1.
The following lemma gives us further insight about the W"* Model Matching Problem. More
specifically, it gives us a lower bound on i, which supports our conjecture that we should invert
T2., away from its imaginary zeros, in the closed right half plane.
Lemma 5.2.1 (Bounds on po)
max{|T1(joo)|, Max|1T1(jok)|,
T,;
}I
p.
:5
||T1||u.
k
Proof
To prove this, we begin by noting that
pO
def
inf lIT 1 - T2 Q||.
QE7W0
< |IT1||l.
since 0 E RO. This proves the upper inequality in the lemma.
To prove the lower inequality, it will suffice to show that each of the terms IT1(joo)I, maxk JT1 (iwk)|,
and JT1 TrT is a lower bound for po.
From the maximum modulus theorem for W** (proposition 2.7.2), it follows that |T1(joo)
11Ti - T2QII. for any Q E 7-O*. It thus follows that
|T1(joo)| 5 pto.
Since T2(jwk) = 0 for each k, from proposition 2.7.2 we have maxk |T1(iwk)|
for any Q E 1o**. It thus follows that
|IT 1 - T20||u.
maxIT1(jwk)|
5 po.
k
|IT1 - T2iT 2.QII. for each Q E Ro.
Since T 2 0 7o* C R*, we have infzEneo |IT 1 - T2iZ||a. J
This then implies that infzEnHo |IT, - T2iZ|lu. < po. By proposition 5.2.1 the left hand side of
r,
this inequality equals 1PT
1
. We thus have
11Tl T2; 1
(Lo.
Combining the above gives us the desired lower inequality:
max{JT1(joo)), maxlT1(jwk),
TT 1T2;
}
j. 0
This completes the proof.
Comment 5.2.3 (Lower bound for p,, at oo)
Note that if we were infimizing over H**, rather than HO*, then we would again have the inequality
IT,(joo)|I pi, if T 2 were to roll-off.
We see that lemma 5.2.1 gives a lower bound on p, which depends on T 2, and on the imaginary
zeros of T 2.. This supports our conjecture that we should invert T2, away from its imaginary zeros,
in the closed right half plane. We shall soon show that the above lower bound is actually equal to
po. The lemma also shows that since T1 E '*, po will necessarily be finite.
Our interest in the "inner problem" will be to use its solution Zo to construct near-optimal
solutions for the 1** Model Matching Problem. The idea will be to appropriately modify ("rolloff") Ti-' Z0 so that the resulting function is admissible 2 and near-optimal. This modification shall
be carried out in two steps. Each step is now described.
Main Ideas: Construction of {Qm,n}.
The first step will be to construct a sequence {fm},=i C N-*
By this we mean that fm will be such that
|IT 1 - T 2iZofMIH.
to modify the "inner solution".
po + 2E
for M sufficiently large. The function fn, as we shall see, will contain all the essential information
about the imaginary axis zeros of T2,. It will be small near each imaginary zero of T2 , and it will
approximate unity elsewhere. To construct the function, we shall exploit the fact that T2 , has a
finite number of zeros on the extended imaginary axis. The phase properties of fm will be critical
in achieving the above inequality.
After constructing the sequence {fm}m* to modify the "inner solution", we shall construct an
inverting sequence {gn}* C RNo for T2,. This terminology will be justified shortly. The sequence
will be such that
T2 1gn E
and
lim |IT 2iZofM(1 - gn)||
n-+oo
oo
= 0
for each M, n E Z+We then define a double sequence according to
Qm,n
def
T2
n
Zoof fm gn.
This sequence will lie in X** for each m, n C Z+. Hence, it will be admissible.
Finally, the above are combined with the following inequality
|IT 1 -
T2QM,nIIoo < IT1 - T2;ZofMIIoo + IIT2ifM(1
-gn
Joo,
to show that there exists M, N E Z+ such that
po
jT 1 - T 2 QM,n hI. < po + 3
for all n > N.
We thus see that, on the one hand, the sequence {f m }* 1 allows us to keep the essential information contained in T2,. On the other hand, the sequence {gn}** allows us to invert that part
of T2, which is not essential. The two sequence together shall implement our idea of invertng T2 ,
away from its imaginary zeros, in the closed right half plane.
The following algebraic result will shed light on the construction of the sequence {f m },
will modify the inner solution.
2In, genieral, T2 , 1 Z, will have poles on the extended imaginary
axis.
1
which
Lemma 5.2.2 (Algebraic Result)
Suppose T,Y, f E ' where If| |1H,. 5 1 and ||Y||1
||T||x.
It then follows that
IT + (Y - T)f12 [8=jW] < max{IT(jw)1 2,|I|Y||.} + 2||T|12{211 - cosu I|+ Isine1 - 6 | +|I |}
(almost) everywhere on the extended imaginary axis.
Proof
The proof of this lemma is purely algebraic and is given to accommodate the skeptical.
IT + (Y - T)f1[28 =j j = IT(1
=
=
2
1T|
(1 -
-
f) + yf|2
T(1 - f)12 + T*(1 - f *)Yf + T(1 - f)Y*f* + |Yf| 2
f - f* + |f|2l) + T*Yf - T*Ylf| 2 + TY*.f* - TY*|f12 + |Yf| 2
= |T12 (1 - 2|f|+ |f12) + 2|T |2 |f| - |T| 2(f + f*) + T*Yf + TY*f* - |f|2 (T*Y + TY*) + |Yf| 2
= |T| 2 (1- If1) 2+2|T| 2 f|(1 -cos)+T*Y(f -f*)+(T*Y+TY*)f*-2|TIYIf| 2 cos(ot-oy)+ |Yf| 2
= |T12(1 _ IfI)2 + 2|T| 2|f|(1 - cosf) + IYf
- 2|T||Y||f|2 cos(Ot - 0Y) + |TI IYIej(** -6)j2|f| sinG
+2|TIY| cos(Ot - Oy)f e-jef
2
After expanding each complex exponential into real and imaginary components, one obtains
IT + (Y - T)f| 2[8=ju] = IT12(l
+2|T12 1fI(1
Ifl)2 + IYfI 2 + 2|TIIYIIfI cos(Ot - O)(1 - IfI)
cos 61) + 2|TI|Y|IfI cos(6t - Oy)(cos6j - 1)
+2|TIIYIIfI sin(Ot - Oy)(sin6f - Of)
-
+2|TIIY|IfI sin(Ot - Gy)65
(almost) everywhere on the extended imaginary axis.
Given this, let
p def max{|T(jw)|,||Y||lj}.
With this definition, and the fact that If||j.
I
1, we then have
IT + (Y - T)f| 2[8 =j]
< A2(1
- 2|fl + If12 + If| 2 + 2|fI - 21f 12)
+4p12 |1 - cos 1 |+ 2pu2 Isin6f - OI + 2p2I8,I
or equivalently,
|T + (Y - T) f| 2[.3 j]
2 +124p2|1
- cos5I| + 2p 2 IsinOf - Of5 + 2y 2
(almost) everywhere on the extended imaginary axis.
Since by assumption IIY|I.
then gives us that
IT + (Y - T )f| 2[
5 ||TH., we have p < ||TH .. This, and the above inequality,
< i2
+ 2||T ||2 {2|1 - cos0u| + |sin65 - Of + |6I}
(almost) everywhere on the extended imaginary axis. This, however, is the desired result.
U
Comment 5.2.4 (A Convexity Argument)
If we let Of = 0 in the above lemma, we get f E [0, 1]. In such a case, the desired inequality follows
from a simple convexity argument:
IT + (Y - T)fV I
(1 - f)ITI + fIY| 5 max{ITI, IYI}.
This simple case captures the essential ingredient needed to construct the sequence {fm}.=1. This
is demonstrated in the following proposition.
Proposition 5.2.2 (Phase Result)
Let {fm},' =1 denote a sequence of ?** functions such that
l|fmI||.
51
for each m E Z+ and
lim IIm|,
= 0.
Given this, there exists M 4! M(E, |IT1||N.) E Z+ (independent of w) such that
IT1 - T2iZofm[,=jI
max{|T 1 (jw)|, IETTr 1} + E
for all m > M, (almost) everywhere on the extended imaginary axis.
Proof
We begin by defining Ytf T1 - T 2 j Zo. This then gives
T1 -T 2 iZofm = T1 + (Y - T1)fm.
From proposition 5.2.1, ||YII
these gives |IYI||.
Tr; . From lemma 5.2.1,
T1 T
rTr
IT1| |.
Combining
5 |IT1||,.,. Given this, from the algebraic result in lemma 5.2.2, it follows that
IT1
-
=
T 2 iZofm| 2
jw] = IT1 + (Y
-
Tl)fm|
2
w
S max{IT1(jW)| 2, lIyI|I.} + 2||T1I.{2|1 - cosO/mI + |sin/m - 6fm +
+6fm|}
everywhere on the extended imaginary axis. Substituting IIYII,. = L"Tr 1 1 into the first
(almost)
term then gives
|T1 - T 2iZofm| 2[M1
max{|T1(jw)|2,T
1
1 r
+2 |12
(almost) everywhere on the extended imaginary axis. From the assumed phase property of fm, it
follows that the last three terms can be made arbitrarily small, uniformly in frequency, by taking m
sufficiently large. More precisely, there exists M =e M(E, IT1 ||Ig..) E Z+ (independent of w) such
that
IT1 - T2Zofm .j]5 max{IT1(jw)I, (ILTT; 1} + E
for all m > M, (almost) everywhere on the extended imaginary axis. This completes the proof. I
We emphasize that only magnitude and phase assumptions on fm were used to obtain the
above result. That such assumptions should suffice follows "intuitively" from the simple convexity
argument given in comment 5.2.4.
In the proof of proposition 5.2.2 above, the phase assumption seemed indespensable. The
following shows that it can sometimes be relaxed.
Comment 5.2.5 (Relaxing the Phase Assumption)
It should be noted that if T1 (jw) = 0, then the phase assumption on fm becomes unnecessary.
Although not transparent from the proofs given, it can be seen as follows. If fml
1 and TI(jw) =
0, then we have
IT1 - T2 jZofmI[=jW] = |YfmI[p=jw]
|YI[p=jw]
||lluyII
IrT1T2
5
'
which proves the claim.
We now construct the sequence {fm},* . It will possess the properties needed to modify the
inner solution Z0 , in the manner suggested earlier. To perform the construction, we exploit the
fact that T2. has a finite number of zeros on the extended imaginary axis.
Proposition 5.2.3 (Sequence to Modify Inner Solution)
There exists a sequence {fm}m=I C 'O of outer functions which possess the following properties:
(1)
(2)
(3)
ciently
fn E Ce for each m E Z+.
||fm|[u
I
1 for each m E Z+The phase of f m can be made arbitrarily small, uniformly in frequency, by taking m suffilarge; i.e.
mim
= 0.
||/m||1,C
(4) fm converges uniformly to unity on all compact frequency intervals excluding the points
i.e. (see definition 2.10.5) for each 6 > 0, however small,
{wk} k;
lim 1(1
-
fm)XR/B(oo,6)UU'
B(wA,6)
O=
0.
(5) If m&> 0, then fm(±jWk) = 0 for each m E Z+Proof
The sequence defined by the following irrational function satisfies conditions (1)-(5) of the
proposition.
def
(m )a)
f==1
s)
a
+--1
S
k
+-
2
2'
M
k
"
"'_L2S
m
We note, in particular, that
2m
for some B E R+. We see that fm is irrational even when the mk are integers. We also note that
fm(s) = [F(s) F- '(s + 1) Gm(s)
This completes the construction.
M
U
-Mffiawvn
-
no"
Comment 5.2.6 (Phase is Critical)
It should be emphasized that it is the phase property of f, that makes it a very special function.
It is easy to construct rational sequences which satisfy the other properties in proposition 5.2.3;
e.g. the sequence defined by
fm(s) c
(
1
(+m
However, finding a rational sequence which also possesses the special phase properties, at first
glance, seems very difficult. Since fm E C,, we know from proposition 2.10.1, that such a sequence
can be constructed.
We now use the sequence {fm},*OI from proposition 5.2.3, in conjunction with proposition 5.2.2,
to obtain the following key theorem. To our knowledge, the key ideas behind the following theorem
first appeared in [14, pp. 68-72].
Theorem 5.2.1 (Modifying the Inner Solution)
Given that T 1 E C. and that T2. has a finite number of zeros on the extended imaginary axis, there
exists M = M(e,T1) E Z+ such that
po + 2E
|IT 1 - T2iZofml 7 .
for all m > M.
Proof
To prove the theorem, we begin by defining
Y = T1 - T 2 Z.
This then gives
Ym
T
- T
2 1 Zofm
=
T
1
+
(Y -
T1)fm.
We shall examine the magnitude of this quantity on two sets. We denote these sets S6 and R/Sb.
To prove the theorem we proceed in three steps. (1) First, we show that there exists a set Sb on
which Im Isatisfies the inequality for m sufficiently large. Here we exploit the continuity of T and
the magnitude and phase properties of f m (modulo proposition 5.2.3). (2) We then exploit the
magnitude and compact convergence properties of fm to show that IYm I will satisfy the inequality
on R/S 6 for m sufficiently large. (3) Finally, we combine the results in steps (1) and (2).
Step 1: Analysis on S6.
In this step we construct a set Sb such that
IYmXsI,1.oo
To do this we proceed as follows.
(a) Constructing Sb.
po + 2E.
=
M"
IiiiiiliC.
First consider the behavior of T1 near joo 3. By assumption 5.2.1, T1 is continuous everywhere on
the extended imaginary axis and hence at joo. It thus follows that there exists f? def
= II(e, T 1 ) E R+
such that IT1(jw) - Ti(joo)| < E for all w such that wI ;> 0; i.e. for each w in B(oo,
n
(-oo, -fl) U (fl, oo). Hence
A)
(T 1 - T1(joo))XB(o,A)
This takes care of the point at oo.
Without loss of generality, we can assume that the points
{wk} o are finite.
Now consider the behavior of T1 near each point
that there exists
6
(-wk - 6k, -Wk +
k 45
6
6
KE.
''
jWk 4. Since T is continuous at jw, it follows
k(E, Ti) E R+ such that |TI(jW) - T1(jwk) < c for all w in B(Wk,6k)
k) U (wk - bk,Wk +
(T
6
k). Hence
< f
- T1(jWk))XB(k,bk) 1
for eack k. This takes care of each point Wk.
We now define
6 =6(e, T1)
=
min{
,mi n 6k}.
This quantity is well defined, and positive, since we are minimizing over a finite number of positive
objects. This is because T2, has a finite number of zeros. Given this, it then follows that
(T
- T1(joo))XB(oo,b)
N
< E
and
(T 1 - T1(iok))XB(wA, )
6
< E
for each k.
We now define the set S6 as follows
Sb tf B(oo, 6) U U_ OB(Wk,6).
This set represents points which are "near" the points {oo, {wk}o}.
(b) Upperbound for T1 on S6.
Here, we would like to obtain an upper bound for T1 on S6 . From part (a), it follows that
T1XB(oo,6)
[Pia5
|ITI(joo)I + J(T1 - T1(joo))XB(o,6) y 5 |T1(joo)i + E.
It also follows that
T1XB(wk,6)J
:5 |T1(jw)J +
(T 1 - T1(jwk))XB(w,s)
5T1(jW)| + E
for each k. Combining the above, then gives us
||T1Xs 6 Io 5
3
4
sup
zE{oo,{wA}kO}
Thie point -oo is treated analogously.
The points -wk are treated analogously.
JT1XB(Z,6)
<
sup
zE{oo,{wA}Q 0 }
|T1(jz)| ±E.
This, however, is equivalent to
||T1Xs,5ue.
max{1T(joo),niax IT1 (jwk), IrTlT;-}+e.
This gives us an upper bound for T1 on S6 . We emphasize that this upper bound was obtained
by exploiting the continuity of T1 and the fact that T 2. only has a finite number of imaginary zeros.
(c) Inequality on S6.
From proposition 5.2.3, we know that there exists M1 = Mi(e, T1 ) E Z+ (independent of w)
such that
IYmgIB.j] max{|T1(jw)|, Tr,
}+ E
for all m> M 1 . In obtaining this, we have exploited the magnitude and phase properties of fm.
Since M 1 is independent of w, the inequality holds (almost) everywhere in S6. Given this, we
obtain ||YmXs, 1,,. 5 max{IITXsII.,
rTrT I} + e for all m> M 1 .
Combining this inequality, with the upper bound obtained above for T1 on S6, gives IYXsllwoo <
max{IT1(joo)I, max ITi(jk)|1,
I } + 2e for all m > M1 .
From lemma 5.2.1, we have max{IT1(joo)|, maxk IT(jwk)|,
have
||Ym X Is,|| : y. + 2e
rT
1r;
}
p.
Given this, we then
for all m > M 1 . This completes the analysis on S6. We now turn to R/S 6 .
Step 2: Analysis on R/S 6 .
We now consider the set R/S 6 =- R/B(oo,6) U U1.=B(wk,b). On this set, for each m E Z+ we
have
Ym XR,
Since |If m ||I,,
=
(Yfm + T1(1 - fm))XR/S,
o
fm
1
(1- fm)XR/s
6
1 for each m E Z+, by proposition 5.2.3, this becomes
III 00 +
1Yu7
YmXR/S,
IIT1 I,1 (1 - fm)XR/S
for each m E Z+
By proposition 5.2.3, the second term can be made arbitrarily small ( 2e) by taking m sufficiently large. How large m needs to be, depends on e, on ||T 1||,., on 6(eT 1 ), and on f(ET).
Consequently, there exists M 2 tf M 2 (E,T1) C Z+ such that
YmXR/S
IIYIIuoo + 2e
for all m > M 2 . In obtaining this, we have exploited the magnitude and compact convergence
propertiesof fm.
From proposition 5.2.1, ||Y|7
=
. From lemma 5.2.1 we have 1Tr
p/. Given
this, we then have
-PTT
YmXR/S,
for all m > M 2 . This completes the analysis on R/S 6 .
Step 3: Combining steps (1) and (2).
o - ZE
Combining the results of (1) and (2) assures the existence of an integer M
max{M1, M 2} E Z+, such that
po + 2E
||Ym1.|7
d'
M(e T1)
cf
for all m > M. This completes the proof.
Comment 5.2.7 (Sifting Property of fm)
Upon inspection of the above proof, we see that
IT1 + (Y - T1)fm||17j 5 max{ IT1(joo)|, maxT1(jok)|, ||YI|y } + 2e
for m sufficiently large. This sifting property of fm, we emphasize, is mainly attributed to the
magnitude and phase properties of fm. Of course, the continuity of T1 is also important.
As we shall see, theorem 5.2.1 above and theorem 5.2.2 below are at the heart of the X"i results
presented in the thesis. Together they will show how one can modify the Z, to the "inner problem"
considered in proposition 5.2.1, in order to construct nearly-optimal solutions to the X" Model
Matching Problem defined in definition 5.2.1.
We now construct the sequence {gn}** 1 . To do so, we exploit the fact the zero structure of T 2.
on the extended imaginary axis.
Proposition 5.2.4 (Inverting Sequence for T 2. )
There exists a sequence {gn}*=1 C
(1) {gn"},
(2) Tij g-
,
.o
of outer functions which possess the following properties:
is uniformly bounded in ?V0.
T 2 .g-1 e V n Ce for each n E Z+.
(3) gn uniformly approximates unity everywhere except on open neighborhoods of the points
{jwk} k=;
i.e. (see definition 2.10.4) for each 6 > 0,
0
lim
fl4o
(4) g'g
n
(1- gn)XR/UlOB(w,5)
kOk
E H**, n Ce for each m,n
00o
= 0.
Z+.
(5) If mk > 0, then gn(wk) = 0 for each n E Z+.
We refer to
Proof
{gnl'" as an inverting sequence for T 20 .
The sequence defined by the function
__mo
def
gn (s) =
1±
_1
(
332 +w
mk
1
mn
satisfies the conditions of the proposition. We emphasize that gn is, in general, irrational since the
Mk, in general, are fractions. We also note that the sequence {gn}' 1 constructed above is the
same inverting sequence that we would use if T2. were real-rational. Finally, it should be noted
that
1
gn(s) = F(s) F-'(a + -) Gn(s).
n
This completes the construction.
Example 5.2.1 (Sample Functions: T20 and gn)
The following indicate that proposition 5.2.4 covers a large class of outer functions T20 .
(a) If 2. =
choose gn =
.
(b) If T 20 = f
choose g
.
(c) If T2. = Q
=
choose gn
.
(d) More complicated outer functions may require the use of [24, pp. 85; theorem 7.4] to
construct gn. The results in [17] may also prove useful.
U
Combining the previous theorem and propositions leads us to the following key theorem.
Theorem 5.2.2 (Near-Optimal Irrational Solution)
Let
def
Qm,n
where Q0
=e
T 2. jZo. Then, Qm,n E
=f
Qofmgn
Ho for each mn E Z+. Also, for each m,n E Z+, we
have Qm,n( jWk) = 0 for each k = 0,1,. .. ,l.
N
-
Moreover, there exists M
=e
M(E,T1) E Z+ and
N(E, M) E Z+ such that
po
|T1 - T2QM,n||Ii.
po + 3
for all n > N.
Finally,from the integer valued functions defined by M(E, T1) and N(E, M), it follows that we
can construct a sequence {Qm}m=1 such that
(1)
Qm E H'
for each m E Z+
(2) If mk > 0, then Qm (jWk) = 0 for each m E Z+-
(3) po < |IT1 - T2Qm||joo < po + 3 for all m > M(E, T1).
Proof
The proof will proceed in three steps.
Step 1: Admissibility of
Qm,,,.
We have
Qm,n 4= T2o'Zofmgn.
By proposition 5.2.3, fm E N* for each m E Z+. By proposition 5.2.4, T 2 j~Ign E H** for each
n E Z+- Combining these gives us that Qm ,n E N* for each m, n E Z+By proposition 5.2.3, fm(jok) = 0 for each m E Z+ and for each k = 1,2,.. .,l. Since
T -1gn E ?** for each n E Z+, it follows that it cannot have poles at the points jwk for any n. This
implies that it cannot cancel the imaginary zeros of fm for any m. Consequently, Qm,n(jo&k) = 0
for each m, n E Z+ and for each k = 0, 1,.. .,l.
Step 2: Near-Optimality of QM,n.
(a) Main Inequality
To prove the rest of the theorem, we use the following inequality:
|IT 1 - T2Qm,nIIj.
= |IT
1
- T 2 iZofmgnI|7 j.
||T 1 - TiZofmI||j +| T 2iZofm(1 - gn)||7..
(b) Use of f m to Modify Inner Solution.
From theorem 5.2.1, |IT 1 - T2jZofmII.
IO + 2c for all m > M =e M(e, T1). Let m = M.
This then gives
/o + 2E + 1IT2iZofM(1 - gn)||0.IT 1 - T2QM,nI 7 o.
(c) Use of gn to Complete Inversion of T 20 .
From proposition 5.2.3, we have that fM C Rw and is zero and continuous at each point jikFrom proposition 5.2.4, gn uniformly approximates unity everywhere except on open neighborhoods
of the points {jwk}1 _; i.e. (see definition 2.10.4) for each 6 > 0,
lim 1(1 - gn )X R/UIn.
(w;,,6)
= 0.
From proposition 5.2.4, the g,, are uniformly bounded. Given this, it follows from lemma 2.10.1
that we can make the last term in the above inequality arbitrarily small by taking n sufficiently
large. More precisely, there exists N t N(e, M) E Z+ such that
||T 2iZofM(1 - 9n1[o < E
for all n > N. This implies that
IT 1 - T2QM,nllHI
< yo + 3E.
for all n > N. This proves the first upper inequality. The lower inequality follows from the definition of p0 and the fact that {QM,n}N
0 '
Step 3: Construction of Qm .
The construction of the sequence {Qm}m"*
follows from the above. This completes the proof. I
Theorems 5.2.1 and 5.2.2 show that under the very weak assumption 5.2.1, we can modify
any solution Zo of the "inner problem" to find a nearly-optimal Q-parameter for the R** Model
Matching Problem defined in definition 5.2.1. Moreover, in the proof of theorems 5.2.1 and 5.2.2
we have indicated how such a Q-parameter can be constructed. The proofs show that to construct
a near-optimal solution one needs to invert T 2, away from its zeros in the closed right half plane.
The following example serves to further illustrate the construction.
Example 5.2.2 (Sample Functions: T 20 , fm, and g,)
Suppose that T2, = (_I)
(a2+1
a+4
The sequence defined by the function f m
satisfies the properties in proposition 5.2.3.
(
1
10m"(
8+1000)
1
(_
The sequence defined by the function g, =
11
\.)
*-j+.OOOl
+-J1
'
i +001
(
*.00
2)
satisfies the
conditions in proposition 5.2.4.
Note:
The sequences {fm}*1 and {gn},"O1 depend only on the zero structure of T2,. They do not not
depend on T 2,.
Although the construction presented in theorem 5.2.2 has been obtained under the weak assumption 5.2.1, the ideas cover a much broader class of T1 and T 2. The following comment is met
to illustrate this point.
Comment 5.2.8 (Form of Construction, Generality, Main Ideas)
In summary, it has been shown that if there exists functions F and G, which satisfy the following
conditions
(1) T2 '(.) F(-) F-'(. + ') Ga(-) E 1*" for each a E Z+,
(2) lim_.o,
F(-) F-1(. +
(3) lime_.o
6F(-) F-1(-+L) Gn(-)
) Gm(-)
K
1,
< B for some B E R+,
(4) The sequence {F(.) F- 1 (. + 1) Gn(-)}L 1 approximates unity on all compact frequency intervals not including an imaginary zero of T24,
then a near-optimal solution to the X** Model Matching Problem can be constructed from the
double sequence defined by the following function
Zo fm gn
Qm,n = T2
where
fm =[ F(s)F-1(s+±)
m
Gm()
m
S+ m
_
and
gd=
[ F(s)
F-1(s
1
+ -)Gn(s) ]
n
Such functions F and Ga were shown to exist under the weak assumption 5.2.1.
The ideas which permitted the above construction shall be used extensively throughout the
thesis. We thus point out the "main ideas".
ru~__-~
_____________
In the construction we modify the "inner solution" Z, by "rolling it off" with f, as in proposition
5.2.3. This really is the main step. It can be done primarily because of the special phase property
of the irrational roll-off function fm.; i.e. the phase of f, can be made arbitrarily small, uniformly
in frequency, by taking m sufficiently large; i.e.
mlim
||6/m Ile = 0.
We then choose m = M sufficiently large. How large M must be chosen depends on e and T1 .
Finally, we show that given M, N can be chosen sufficently large so that
QM,N
=
T20 'ZofMgN(M)
is near-optimal. How large N must be depends on e and M. Here the main role of g, is to assure
the strict propriety of QM,N and to assure that it has no poles on the imaginary axis. It allows us
to invert T2, in a stable manner.
Thus far we have only addressed the problem of constructing near-optimal solutions for the
'H** Model Matching Problem defined in definition 5.2.1. We now address the computation of the
quantity p.
5.3
Computation of po
Often it is necessary to determine y,. The following proposition indicates how one might, in
priciple, compute the value p,,. All assumptions made in the previous section are assumed to hold
here.
Proposition 5.3.1 Each of the following equal p,:
inf0 ||T1 - T2Q||u-
(1)
QEWO
|IT 1 - T2jQ||u.
inf
11EN**
(2)
{Q(jok)=O; k=0,1,...,l}
max{ IT1(joo)|, maxIT1(jwk)|,
Proof
rTr
}
The proof shall proceed as follows: (a) 2=3 (b) 1=3.
(a) (2=3)
We claim that the key to proving proposition 5.3.1 is showing that (2=3); i.e.
inf
|IT 1 - T2jQ||j
=p
{Q(jwA)=O; k=0,1,---,1}
where
pmax{|T1(joo)|, max|T1(jok)|,
rr
.,
(3)
We know from proposition 5.2.1 that infzeioo |IT 1 - T2 iZIJ|u = 07,T
which satisfy Q(jwk) = 0 for each k also lie in ?**, we have
||T
=
r l T,
inf " |IT 1 - T2 iZ||u
ZE
inf
<
|IT 1
QE N0
.
Since the set of Q E X0
T2iQII||W.
-
{Q(jwk)=O; k=0,1,...,}
|IT 1 - T2iQII.
From proposition 2.7.2 we have |Ti(joo)|
Consequently,
k.
each
0
for
=
Q(jwk)
satisfy
which
|IT 1 - T2iQI||uO.
inf
IT1(joo)
for each Q E R**, including those
{Q(jwk)=o; k=0,1,...,4}
For each Q E XH0* which satisfies Q(jok) = 0 for each k, proposition 2.7.2 gives us ITl(iwk)I
|IT 1 - T2jQ||u. Consequently,
I
max|IT1(jok)|
k
inf
<
||T1 - T2iQ||Ieo
QEng
{Q(jwk)=O; k=O,...,l}
Combining the above then gives
def
=
max{IT1(joo)I,maxlT(jwk)|,||YI|,o} <
inf
k
QEnO
|IT 1 - T2iQ Ilu.
{Q(iwk)=O; k=O,...,1}
To show that (2=3), we thus only need to prove the converse inequality. To prove the converse
inequality, it suffices to construct a sequence {Zmm}*=1 C ** such that Zm(jok) = 0 for each
k = 0, 1,..., 1, and such that
|IT 1 - T2iZmI|o. 5 max{IT1(joo)I,maxIT1(Wk)|, JTr;}
+ 3e
for m sufficiently large. The existence of such a sequence, however, is guaranteed by theorem 5.2.1;
just choose Zm = Zofm. This proves the converse inequality and hence that (2=3).
(b) (1=3)
We now prove that (1=3); i.e.
po
def
-
inf ||T1 -T2QI||u.
= P,
where again
def
=
max{|T(joo)I,max|T1(j
),iWk)IrT }
From lemma 5.2.1, we have that
P ! po.
To prove that (1=3) we need to prove the converse inequality. To do so it suffices to construct
a sequence {Q m
1 C ROO such that
lm""=
IT1 - T2Qm|Iu-
max{IT1(joo)I, max| T1(jWk)|, F T
}+3e
for m sufficiently large. The existence of such a sequence, however, is guaranteed by theorem 5.2.2;
just choose Qm = T 2 j'Zofmg, where m = M(E,T1) and n = N(E,T1) as defined in the theorem.
This proves the converse inequality and hence that (1=3).
Proposition 5.3.1 shows that
yo = max{ITi(joo)I, maxITi(jo)|, TPT 1 T2*Given this, we have the following corollary.
Corollary 5.3.1 (Spectral Result)
If
max{IT1(joo)I, maxIT1(jwk)I}
rTiT
,
then
YO =
PT1 T2*;
-
This result shows that if the assumption in corollary 5.3.1 is satisfied, then to compute pA all
that we need do is compute rT 1T. ; the operator norm of the Hankel operator rTTr. This,
in general, involves solving an infinite-dimensional eigenvalue/eigenfunction problem [61]. This
computation can be particularly difficult when the operator is not compact. In the sequel we will
show how this computation, in many cases (including the non-compact case), may be carried out
by solving a sequence of finite-dimensional eigenvalue/eigenvector problems instead.
The following lemma is presented to show how one might construct T1 so that the assumption
of corollary 5.3.1 is satisfied.
Lemma 5.3.1 (Construction of T1 )
Suppose that T71 E X1 B and satisfies
max{IT1(joo)|, maxIT1(jwk)|} = inf ITi(jw)I.
k
wERe
Also, let z be any point in the extended open right half plane, such that T 2 (z) = 0. We then have
max{IT1(joo)I,max|T1(jok)|}
Proof
From proposition 5.2.1, there exists Zo E 1*
T
T2
such that
lI,T1 - T2 ZoIIw-
From proposition 2.7.2, we have
2 IT1(z)l
Since Tj' E XIB, the minimum modulus theorem [50, pp. 159] can be applied to get
> inf |T1(jo)|.
WER,
The result then follows since the right hand side equals max{IT1(joo)|, maxk |T1(jwk)I} by construction.
This lemma can be applied to T 2 j = e-8, for example.
5.4
Computation of Hankel Norm
The difficulty in computing p, can be attributed to the difficulty in computing the Hankel norm
rTTi . This computation typically requires that one solve an infinite dimensional eigenvalue/eigenfunction
problem [14, pp. 28-31], [61], [63, pp. 308]. In this section, it is shown how the computation can
be carried out by solving a sequence of finite dimensional eigenvalue/eigenvector problems.
Throughout this section we assume that {T 1 .} 1,{T 2,n}0* C RX"O, where T 2,, is inner in
1" for all n E Z+. We begin with the following proposition. Loosely speaking, it shows that
ITi1 is lower-semicontinuous in T2 in the compact topology on H".
Proposition 5.4.1 ("Lower-sernicontinuity" Hankel Result)
Suppose that
liM |iTI
-
T1||I
=0
and
lim1 (T 2 n
= 0
T 2 j)X[-n,O
-
for each 0 E R+, however large. It then follows that
Trr
T
Proof
Let v E C2 be such that
1v1LC2
lim
=
rT,
.
1 and
rTJ;
TT
2.
Vr
2 +E.
Given this, it follows that
11rTS 1:
~T1 T;
1T(2
-Tni +2.)s£j2
PT(T 2-T.+Ta
2 )*V(
<
1T1(T2 i
+
+£2E
< 1rT, (T 2 jT
7-v
-T2 )*vI1, +
2
ni).V
VTT*
£C2
+
+
PT~r1 T2*ivC
.
Since v is not necessarily a "maximal vector" for LTT; , it follows that
1T21
T1(T2
-T2)*Vf
+ ITT
I+E
From lemma 2.10.2 and our assumption on T 2 ., it follows that the first term can be made arbitrarily
small by taking n sufficiently large. Consequently, there exists N t N(E, T 1, v) such that
11TT*T2.- 11:
T 1 T2n
I±
2,E
for all n > N. We also have
r T,T*
-
T
;
+
T
:5 rT T;
Combining this inequality with the previous, gives the result.
+ ||T 1 - T16||0.
U
Since T1 E C,, we know from proposition 2.10.1 that there exists a sequence of RH" functions
{T 1.}'* 1 which uniformly approximate T1 . Given this proposition, we have the following theorem
which gives insight into the computation of T1 T,2 ..
Theorem 5.4.1 (Spectral Result: Computation of Hankel Norms)
Given that
lim ||TI, - T1 ||%. = 0,
fl--+0
each of the following imply that limn_.o
=
rT 1 ,;
lim T2 n. - T2
n--+oo
=0
lim
(2)
0
7joo
(T 2 n. - T 2i)X[-n,O]
(3)
= 0
(4)
= 0
- T 2i)X[O,O]
(T2
lim
TTrT.;
|Ti(joo)|
(1)
'7.to0
T1(T 2n. - TO2)
lim n--+oo
T1(joo) = 0;
.
rTT
for each fl G R+, however large.
I
Proof
The proof of (1) - (3) follows from the following inequality
- IrT1T2.
T1,T
< Ir(T -T)T*
|5
T1lTn
||T1
T(Tn -T')
-T1T-
+ IT1(T 2
T1||Io.
.
-
T2 )
-
To prove (4), we invoke proposition 5.4.1 to obtain
K
'T1T*
lim
T1
2Tnj
This proves one direction. To prove the other direction we proceed as follows. Let Q0 E ?10 be
such that
IT, - T2jQ11ue
max{ |T1(joo)|,
rTr
}T+.
The existence of such a Q0 is guaranteed by theorem 5.2.1. Given this, we have
rTnT
1T1, - T 2 i Z
=
|IT 1 - T2 Q0||u. + |IT1, - T1I| 7
< max{ IT(joo)l,
By assumption, IT,(joo)|I
L'T1 T;
rTrT;
rT
1;
}
+
E+
:o 1T1, - T2ni Qo
o
+ (T2i - T2 . )Qo
||T1, - T1%. + I(T 2i - T2, )QO
-
. Hence, the above inequality becomes
E+
IT1, - T1|II
+ (T21 - T 2,.)Qo
The second norm can be made arbitrarily small by taking n sufficiently large. This follows
by our assumption that TI, approximates T, uniformly. By lemma 2.10.1, the last term can be
76
made arbitrarily small by taking n sufficiently large. This is because T 2 ,,. is uniformly bounded,
it approximates T2, on compact frequency intervals, and Q, rolls-off. Consequently, there exists
N = N(e, Q,) E Z+ such that
rT 1 T2
T
P1rT;T2
+ 3E
for all n > N. This proves the other direction and completes the proof.
I
Comment 5.4.1 (Applicability, Spectral Implications)
We note that the criterion in (1) is a continuity statement. It implies that T, 1; is continuous
in T2, when T1i uniformly approximates T1 . Since RR"0 is not dense in N**, the criterion in (1) is
usually not applicable; e.g. T 2, = e-' cannot be approximated uniformly by RX"'> functions.
The criteria in (2) and (3) are helpful only when T1 rolls-off.
Unlike the other criterion, the criterion in (4) is usually applicable. It applies to cases in which
the Hankel operator ITT2; is non-compact; e.g. T1 E RX** proper and T2, = e-' (cf. proposition
2.9.2). In such cases, the non-compact Hankel operator cannot be approximated by finite rank
operators. Moreover, it is not clear to us how one would approximate a non-compact Hankel
operator 5 . It also applies in instances where T2 j has an infinite number of poles and/or zeros;
e.g. an infinite Blaschke product. To verify the inequality criterion in (4), lemma 5.3.1 may prove
helpful. If oo is an essential singularity of T2j, then the inequality is automatically satisfied. This
has been shown in [61].
Finally, the spectral implications of theorem 5.4.1 must be acknowlegded. More specifically, we
reiterate that computing 'Tr 1 amounts to solving an infinite dimensional eigenvalue/eigenfunction
problem [14, pp. 28-31], [61], [63, pp. 308]. Theorem 5.4.1 gives weak conditions under which such
a computation can be carried out by solving a sequence of finite dimensional eigenvalue-eigenvector
problems [23]. Since computing the above Hankel norm is crucial in many X** design procedures,
this theorem is extremely useful.
U
5.5
Sequences of Finite Dimensional 70 Model Matching Problems
The R** Model Matching Problem, in general, defines an infinite dimensional optimization problem.
Obtaining a solution is often difficult and requires sophisticated mathematics 6. It is thus natural
to ask whether or not solving a sequence of appropriately formulated finite dimensional problems
can yield desirable results. Solving a finite dimensional Ri0* Model Matching Problem requires little
mathematical sophistication. Also, much software exists to support such an approach.
Let {T1.}*_1 and {T 2 .}**n be sequences of RX** functions. Consider the finite dimensional
R** Model Matching Problem defined by the following optimization problem.
Definition 5.5.1
.n
=
inf
QERWiO
||T1. - T2nQ|I
'Distribution theory may be helpful in answering this question.
6
Computing Z0 may be difficult.
Our goal is to gain insight into the infinite dimensional problem defined in definition 5.2.1, by
studying the finite dimensional model matching problem posed in definition 5.5.1. This, of course,
is reasonable only if T1 , and T2, approximate T1 and T2, in some sense. We thus make additional
assumptions on the sequences {T 1,}** 1 and {T2n}*1
Assumption 5.5.1 (T1 , Approximates T1 Uniformly)
Throughout the section, it shall be assumed that the sequence {TI,}** 1 satisfies
lirM |T1, - T1||I
n-.#oo
= 0.
Since T 1 E Ce, we know from proposition 2.10.1 that it can be uniformly approximated by R1**
functions. Hence, this assumption is justified.
As with the X** Model Matching Problem, the above finite dimensional model matching problem
must be approached with an inversion mentality. Given this, it should be clear, for example, that
we will need to invert the outer part T 2 no of T 2 ., away from its imaginary zeros, in the closed right
half plane. This statement has already been implemented for T2 , in section 5.2. To do the same for
T2 n, we will need to construct T2 in a clever way. It should be apparent, for example, that choosing
T2n so that it uniformly approximates T2, in X**, will not suffice. This is because T 2no and T2.,
even for large n, may possess drastically different zero structures on the extended imaginary axis.
We now show how to prevent this from occurring.
From proposition 5.2.4, we know that there exists a sequence {gN}N= 1 C Rw of outer functions
such that
(1) {gN}N=1 is uniformly bounded in XH.
(2) T; 'gN , T2 ,gN 1 E R
Cee for each N E Z+(3) gN uniformly approximates unity everywhere except on open neighborhoods of the points
{jWk}'k= 0 ; i.e. (see definition 2.10.4) for each 6> 0,
lim
N--*oo
(1
-
gN)XR/U
B(wh,6)
'
=
0.
(4) g ggN E 7** n Ce for each M, N E Z+(5) If Mk > 0, then gN(Wk) = 0 for each N E Z+The above properties were critical in "inverting" T 2.. We referred to {gN}N=1 as an inverting
sequence for T 2.. The following proposition, we shall see, will guarantee the existence of such a
sequence for T 2n.; i.e. provided that T2, is constructed appropriately.
Proposition 5.5.1 (Inverting Sequence for T n )
2 0
Fix N E Z+. There exists a sequence {n}' 1 C RX** of outer functions (which depend on N)
with the following properties:
(1) limn_.oo |n - gNI ,H = 0(2) For each A E Z+,
lim |IfA(1 - gn)||g., = 0.
(3)
-§n
E 1** n Ce for each m,n E Z+-
(4) The only imaginary zeros of gn are the points {wk}k=o; each with multiplicity 1.
We shall refer to {n}'
Ptoof
1
as an inverting sequence for T2 ., which we shall be construct below.
The proof follows by inspection of the function
def
N
a
s+k
mo
2
-92
m
1(2+2Ns+wl+12+
N"
and the use of proposition 2.10.1. The idea is to factor out the essential information (i.e. the
imaginary zeros) and approximate the rest by minimum phase RW* functions.
I
We now show how to construct the sequence {T 2 n} -1 so that it approximates T 2 in a manner
which makes the associated finite dimensional model matching problem an appropriate tool for
studying the infinite dimensional problem posed in definition 5.2.1. The construction is such that
for large n, T 2no contains all the "essential information". More specifically, T 2,. will approximate
T2, uniformly on compact frequency intervals, T2 n will approximate T2, uniformly, and {n}I
will
be an inverting sequence for T2 ,,. The construction presented shall be exploited in later chapters
on c* design.
Construction 5.5.1 (Construction of {T 2 }*- 1 )
Let {T2,}1 0 C R2'
be any inner sequence such that
lim1 (T 2 n. - T 2 j)X[-o,n]
=
0
for each 11 E R+, however large. Let {An} ' 1 C R -* be any minimum phase sequence such that
lim An - T2,gJ
n--*oo
11
= 0.
Here, N can be any positive integer. We then define
T 2n0 L A=n
and
def
T2.= T 2 noT 2 n.
Given the above, we have that {T 2.}*=
following
1
is a uniformly bounded sequence which satisfies the
lim |IT2no - T2 0||,
sup T22'5B
n
7o
=0
(1)
< MB < 00
(2)
for each B E Z+ and
lim (T 2.
for each f E R+.
-
T2 )X[_O,O]1
=0
(3)
Proof
(1) follows from the following inequality
||T2o - T2*1||wo
= ||#n An - T2.||wo
:5
||An||wjo
||9n -
NI17oo +
19N Noo-
An -
T2,9gN1
and proposition 5.5.1. We note that (1) implies that the sequence {T 2 0 *0 is uniformly bounded.
Since T2n is inner, it also follows that the sequence {T2}**, is also uniformly bounded.
By construction, T2.gg' is bounded away from zero (cf. proposition 5.2.4). Moreover, it is
minimum phase. Consequently, by proposition 2.10.1, there exists a minimum phase sequence
{An}* 1 which uniformly approximates T2.gg'. Since An converges uniformly to T29N- 1 , it
follows that An will be bounded away from zero for sufficiently large n. Given this, we obtain (2)
from
T 2n
<
N1HOT
B
N
11_
JAn-
Bjj,
11
and proposition 5.5.1.
Finally, (3) follows from (1), the following inequality
(T2, - T 2 )X
0
,, 1g11
| T2
(T2n,
and the fact that the sequence {T2n}*
-
T2)X -0,nj
+ |jT 2 j(T 2,
-
T2.)||j.,
is uniformly bounded.
Comment 5.5.1 (Practicality)
The condition on the sequence {T2.}
is reasonable. Inner functions in 1H** can "usually" be
approximated on compact subsets, by functions in RW**. If T2j is a delay, for example, one can
use Pade' approximations to obtain the approximants T2n.. See example 2.10.2.
I
The following proposition, loosely speaking, shows that, for large n, y, is bounded from above
by p,,. It should be interpreted as an upper-semicontinuity result (53, pp. 345].
Proposition 5.5.2 (Upper-semicontinuity)
lim pn < po.
fl-+00
Proof
The proof of this proposition follows from the following inequality
pin
IIT1,
-
T2.QOll.
!5 ||T1 -
T2QOII.
+ ||T1, - T1||-
+ ||(T2 -T2.)QOllWy
which holds for each Qo E R**. By definition of po, there exists Qo E R** such that |IT1 - T2Qoll <
pO + E. This takes care of the first term on the right hand side of the inequality. The second term
can be made arbitraily large by taking n sufficiently large. This is because T1, uniformly approximates T1. The third term can also be made arbitrarily small by taking n sufficiently large. This
follows becase T2, is a compact approximant for T2 and because Q0 rolls-off. See lemma 2.10.1.
Consequently, there exists N f N(E, T1, T2) E Z+ such that
lim00 pn 5
fl-*
Io +
3E
for all n > N. This, however, proves the result.
Comment 5.5.2 (Uniform Boundedness)
Since p, 5 |IT1||7oo, we know that p,, is bounded. Given this, the above proposition implies that
and T1, uniformly
ptn is uniformly bounded. This is also known from the fact that pn < IIT, 1,.,
approximates T1 .
The following theorem shows how to construct near-optimal solutions for the finite dimensional
model matching problem defined by definition 5.5.1.
Theorem 5.5.1 (A "Sub-Optimal" Real-Rational Sequence)
There exists a sequence
such that
{Qn}
E RtoO which is uniformly bounded, uniformly rolls-off, and
Ti, - T 2nQn
pn
I
pn
+
E
for n sufficiently large.
Proof
From proposition 5.2.1, there exists Zn E RHOO such that
||Yn||yo =
-u0
T 2 i.Z
where Yn =- T 1 , - T 2 n Zn. Moreover such a Zn is easy to compute since T1 , and T 2 , are real-rational
[23]. Now define
def
Qab,n - T'
Znfa#b
where fa is as in proposition 5.2.3, #b is as in proposition 5.5.1. We first show that Qa,b,n is nearoptimal for appropriate choice of a and b.
Step 1: Near-Optimality of QA,B,n.
Consider the following inequality.
IT1n - T2,Qa,b,n|hoo = T, - T 2n Zn fa~b
IT1n + (Yn
< |IT 1 + (Yn - TI)fa||eoo + IT1,
5 Tn - T2n Znfa
+ T 2 n Znfa(l- #b)
T1.)faII|o + T 2 niZnfa(1 - #b)
-
-
Til.
+ II(T1 - TIn)fal|oo + JT 2 n Znfa(1
-
#b)
-
From theorem 5.2.1, we know that there exists A t= A(E, T1 ) E Z+ (independent of n )such that
IT1 + (Yn
for all n E Z+.
- T1)fA||I-
I|YnIIoo} + f
5 max{IT1(joo)|, maxIT1(jWk)|,
k
By construction
pu, = max{|T1.(joo)|,mnax|T1.(jos)J,||tYnIIu.}.
k
Since T1 . uniformly approximates T 1 , there exists N1
=e
N1 (E,T1 ) E Z+ such that
max{IT1(joo)I, maxIT1(jWk)I,
I|Y|1nII}
k
An + E
for all n > N 1. Given this, the previous inequality becomes
||T + (Yn -T1)fA||o
jPn+E
for all n > N 1. From the first inequality, this then gives
||T1. - T2nQA,b,nll.. :5 An + E+ ||TIn - TiI.
+ II(T1 - T1.)fAIIu
+ T2 .i ZnfA(1 - 9b)
-
The second and third norms on the right hand side can be made arbitrarily small by taking n
sufficiently large. This follows because T 1. uniformly approximates T1 and because fA is a fixed
H 0* function.
To make the last term arbitrarily small, we begin by noting that T2n. Z is uniformly bounded
in W**. This follows from the inequality
T 2 ni Zn11
zin
WO ZER71-
Ti
+ IIT1nIIK.
- T 2 n.Z
2|IT1.|II,
and the fact that T 1. is uniformly bounded. Given this, proposition 5.5.1 implies that b can
be chosen sufficiently large to make the last term arbitrarily small. Cosequently, there exists
B =f B(E, A) E Z+ (independent of n) such that
<
T2,iZnfA(1 - B)
_
for all n E Z+.
Combining the above, we have that there exists N > N1 in Z+ such that
I|Ti.
- T2nQA,B,nI|.o
!5 An
+ 4E
for all n > N. This shows that QA,B,n is near-optimal for large sufficiently large n.
Step 2: Uniform Boundedness.
We now show that {QA,B,}"l 1 is uniformly bounded in X**. We obsernve that ZnfA is
uniformly bounded since Zn is and since fA is a fixed R*O function. We then note that
sup T2
NB
! MB
<
for each B E Z+, from construction 5.5.1. These imply the uniform boundedness of {QA,B,n}"
1.
Step 3: Uniform Roll-off.
That {Q AB} 1 is also uniformly rolls-off follows since T'
since fA rolls-off (cf. proposition 5.2.3).
Zn§B is uniformly bounded and
Step 4: Construction of Qn.
Finally, we note that fA is the only irrational function used in constructing QA,B,n. However,
by proposition 5.2.3, fA E Ce. Hence, from proposition 2.10.1, fA can be uniformly approximated
by RVI* functions which possess similar properties. The construction of the sequence {Q71}n*
thus follows.
I
The following theorem guarantees the converse of proposition 5.5.2. It shows that pn approaches
po as the approximants get better.
Theorem 5.5.2 ("Continuity" Result)
lim pn = p1.
Proof
Proposition 5.5.2 gives upper-semicontinuity:
lim pn
p.
Yo
n-.+oo
We now prove lower-semicontinuity. We begin with the following inequality
p: SITi - T20n
<T
1
. -T
2
Qn J
+
IT1 - T1. 1||w +
(T2 . - T2)Qn7
where Qn is constructed as in theorem 5.5.1.
The first term can be made arbitrarily close to pn by constructing e, appropriately.
The second term can be made arbitrarily small by taking n sufficently large. This is because
T1 . uniformly approximates T 1 .
Finally, the third term can be made arbitrarily large by taking n sufficiently large. This is
because T 2 . is a compact approximant for T2 (cf. construction 5.5.1) and because Qn is uniformly
bounded and uniformly rolls-off (cf. lemma 2.10.1).
Consequently,
pto
lim pn
n--+oo
for n sufficiently large. This proves the converse inequality (lower-semicontinuity), and hence the
theorem.
g
Comment 5.5.3 (The case T1. = T1 )
Upon inspection of the above proof, we see that if T1 . = T1 , then the above result would hold for
all n E Z+. Otherwise, the result is only guaranteed for sufficiently large n. This is due to the way
in which we construct Qn. It should be apparent that a different construction could be guaranteed
to be near-optimal for all n. However, it is not obvious that such a construction would be uniformly
bounded and would uniformly roll-off.
F
From the above theorems we get the following corollary.
Corollary 5.5.1 (A Near-Optimal Real-Rational Solution)
There exists a seqence {Ql},*_1 E RR* which is uniformly bounded, uniformly rolls-off, and
such that
T1 - T 2Qn
!
po
< po
+e
for each n sufficiently large.
Here we take {Q}nl* 1 as constructed in theorem 5.5.1. The lower inequality is obvious
Proof
since On E R1** and is thus admissible.
The prove the upper inequality, we consider the following inequality
T1 - T2 Qnjj,,
5 T1 ,
-
T 2 Qn
+ 1IT 1 - T1.||WI
+ (T2
-
T2 .)Qn
.
We then note that that the last two terms can be made arbitrarily small by taking n sufficiently
large. The result then follows from the fact that
1T1,n-
T2,Qn 110
<pn +61,
for n sufficiently large, and theorem 5.5.2.
5.6
Summary
In this chapter we defined the R1** Model Matching Problem and showed how to construct nearoptimal solutions for it. Sequences of appropriately formulated finite dimensional X** Model Matching Problems were also considered in an effort to avoid the complex infinite dimensional N** Model
Matching Problem. It was shown that near-optimal real-rational solutions could be constructed for
the infinite dimensional R** Model Matching Problem from those resulting from the finite dimensional problems. The ideas presented in the construction shall be heavily exploited in subsequent
chapters on N** design.
Chapter 6
Design via 7o( Sensitivity
Optimization
6.1
Introduction
In this chapter we consider the problem of designing near-optimal finite dimensional compensators
for infinite dimensional plants via R" sensitivity optimization. Such an approach can be motivated by design specifications which require some specified degree of robustness or C2 disturbance
rejection. A systematic procedure is presented. More specifcally, we provide a solution to the R**1
Approximate/Design Sensitivity Problem, the RO* Purely Finite Dimensional Sensitivity Problem,
and the R** Loop Convergence Sensitivity Problem.
6.2
7
Approximate/Design Sensitivity Problem
In this section we present some definitions and assumptions to precisely state the R" Approximate/Design Sensitivity Problem. Notation to be used throughout the chapter is also established.
Throughout the chapter we shall be working over the ring R**. A transfer function will be
called stable if and only if it belongs to R1**. Given this, F(X*) will denote the fraction field
associated with R**. Fc(H**) will denote those elements of F(Oc*) which can be represented as
the ratio of coprime factors in R".
It has been shown in [52] that Fe(R**) consists of all elements in F(R**) which are stabilizable.
With this in mind, we make the following assumption about the infinite dimensional plant P.
Assumption 6.2.1 (Permissible Plants)
P E Fe(H ) 1-
This assumption permits us to exploit the algebraic ideas presented in chapter 3. More specifically, the assumption allows us to associate a coprime factorization (Np, D,) E R** for P, over
R**, with Bezout factors (Nk, D&) E R**. That is,
P =
N
-s-
D,
'Although not explicitly stated, the causality of P is assumed.
and
DDk
-
NpNk = 1.
From proposition 3.2.1, it then follows that the set of all compensators which internally stabilize
P, with respect to the ring '**, are parameterized by
K(P,Q) (L-eNk - DpQ
Dk - NQ
where
Q is any
element in R*.
The following assumption is made regarding Nk and Dk.
Assumption 6.2.2 (Nominal Compensator)
(1) Nk E ?i** n Ce.
(2) Dk E Ce.
(3) Both Nk and Dk are known.
I
Comment 6.2.1 (Practicality, Computational Issues)
Since we ultimately want a near-optimal strictly proper compensator, it makes sense to let the
nominal compensator Kno,, '
be strictly proper. Having Nk E 7o* is thus justified. The
continuity of Nk and Dk is needed so that they mey be approximated uniformly by RX** functions
(cf. proposition 2.10.1). Assumption 6.2.2 is thus made with little loss of generality. The fact that
we need to know the pair N and DA is not very demanding. Since we seek a near-optimal strictly
proper compensator, we should be able to come up with one strictly proper stabilizing compensator.
We do acknowlegde that, in general, the computation of N,, D,, NA, and Dk may be difficult.
This issue shall not be addressed in this thesis.
Given that Nk E 11*, it follows that if we allow Q to vary over X1*, then we get all strictly
proper compensators which internally stabilize P. We shall be doing this throughout the section;
i.e. all infimizations involving P shall be carried out over X**.
In this chapter we shall construct a sequence of real-rational approximants Pn for P. They
will be constructed such that they are elements of Fe(R"). Given this, we are guaranteed the
existence of a coprime factorization (N,, D,) E RX** for Pn, over RW**, with Bezout factors
(Nkn., Dk.) E RR** [56]. That is,
Pn =
Nn
Dn
and
(D,,Dk, - N,.Nk.)-' E RR*.
The functions Pn, N, D,,, Nkn, and Dk. shall be constructed below. From proposition 3.2.1, it
then follows that the set of all compensators which internally stabilize Pn, with respect to the ring
R**, are parameterized by
K(PnQ) def Nk. - D,,Q
Dk, - N,.Q
where
Q is any
element in
*".
The function Nk, will be constructed so that it rolls-off. Given this, it follows that if we allow
Q to vary over R1**, then we get all strictly proper compensators which internally stabilize P. We
shall be doing this throughout the section; i.e. all infimizations which involve P, shall be carried
out over R1*
Throughout the chapter, we shall assume the following inner-outer factorizations over H**:
N, = Ni N,,
N,,
=N,
D,
Np,,
D,=
=
DpD
D,, Dp,.
Such factorizations are guaranteed to exist by proposition 2.7.6.
Comment 6.2.2 (Computational Issues)
It should be noted that the computation of inner-outer factorizations for N, and D, may be
difficult in practice. Such factorizations, however, can be obtained for a large class of plants. The
assumption that inner-outer factorizations for N, and D, are known is thus justified. This issue
shall not receive further consideration in this thesis.
In this section we shall formulate an ?** weighted sensitivity problem. To do so, we shall
require a frequency dependent weighting function W. The following assumption shall be made on
W.
Assumption 6.2.3 (Weighting Function)
(1) W E RH*.
(2) W is outer.
Here, W may be proper or strictly proper.
We now define the notion of an H1o* -sensitivity measure as follows.
Definition 6.2.1 (nX* -Sensitivity Measure)
Let Q E 11* and F, G, E h(H1**). Also, let K(G, Q) denote a compensator which internally
stabilizes G with respect to the ring H*. If it also internally stabilizes F, it is appropriate to
define the H** -sensitivity measure of the pair (F, K(G, Q)) as follows:
W
Juo. (F, K(G, Q)) def
=
1 - FK(G, Q)no
From definition 8.2.1 above, it follows that Juoo (P, K(P,Q))
into definition 4.3.1 and allowing
Q to vary over
f
1H*, then gives us the following expression for the
optimal performance, yvt.
Definition 6.2.2 (Optimal Performance)
wTQI
Winf
pot def inf
Ptu=
where T 1
cef
1 - PK(P,Q)
WDDk and T2 L WDNp.
Substituting
|T1 - T2Q||uco
HQEQEur
U
We emphasize that this definition defines an infinite dimensional optimization problem; one which
we want to, and will, avoid solving. Moreover, we note that the problem is an 7** Model Matching
Problem, such as the one studied in section 5.2.
Similarly, from definition 8.2.1, it follows that Jo- (Pn, K(Pn, Q))
.
After
substituting this into definition 5.5.1, and allowing Q to vary over RH*, one obtains the following
expression for the expected performance, pn.
Definition 6.2.3 (Expected Performance)
defdW
pn =
where T1 ,t
=
inf
Q6 Ru0
1 - PK(
WD,,Dk, and T2,
=f
Q)
P,
,,
=
inf
QE Ru
||T1 - T2.Q||W
-0"
WD, N
Here we can infimize over RX8* since W and P are real-rational [23]. We note that this definition
defines a sequence of finite dimensional model matching problems.
An optimal or near-optimal solution to this problem is typically found by first considering the
"inner problem" [23]:
inf
-lc
ZE Ru*H|
T1, - T2,
Z|
.
Here, T 2 n = D , 1 N,
is the inner part of T2,
f WD,tNP,
(recall that W is outer).
proposition 5.2.1, we know that there exists Z E R**such that
T1, - T2 i ZLi
= zeN
From
T 1, - T 2 , Z 11
Moreover, Zn is unique [23]. Lets define
Q nte- T2 -'
where T2n = WD,, N,
sator
is the outer part of T 2 .. This
Kn
def K(Pn,
Q,
generates a finite dimensional compen-
- DpnQn
Dk, - NPQn'
Qn) = Nk,
This compensator, as we shall see (section 6.3), need not even stabilize P. We thus need to modify
it. For this reason, we define a roll-off operator
r :
Qn -->+Qn
E RXH*.
To construct r we shall exploit theorem 5.5.1. The exact form of r will be determined subsequently.
The compensator generated by Q is given by
kn tefK(Pn, QO) = Nk, - DPnQn
D,
- NPnQn
Given this, we consider the feedback system obtained by substituting kn into a closed loop
system with the infinite dimensional plant P. Let H(P,kn) denote the resulting closed loop
transfer function matrix from r, d to e, u. We then have
H(P, kn)
where
Il
H(P,k,)=
[
P]
'n1A
1
P
Substituting for Kn, then gives
H(P,kn(n))
=
D,(Dk
Dp(Nk
-
-
N,,Qn)
Np(Dk, - N,,Qn)
- NpQn) I
D,,Qn) D,(Dk,
6(P,k(O))
where
6(P,k((Q))
=
D,(Ds, - N,,Qn) - N,(Nk, - D,,Qn).
Given that internal stability can be shown, the actual performance, An defined in definition
4.3.3 is well defined and becomes:
Definition 6.2.4 (Actual Performance)
d
Aef11
W
- Pkn 17o.
_
W(D, - D,,)(Dk, - N,,Qn) + (TIn - T 2 .Qn)
6(P, kn(Qn)7j0
Given the above definitions, the R** Approximate/Design Sensitivity Problem then becomes to
find conditions on the approximants {Pn}**, and on the roll-off operatorr, such that the actual
performanceapproaches the optimal performance; i.e.
lim [,i = pt.
Equivalently, this problem can be viewed as that of finding a near-optimal compensator for the
infinite dimensional plant P. The problem also addresses the question: What is a "good" finite
dimensional approximant?
Because this problem is of primary concern in this research, we now indicate what difficulties
are associated with the problem.
6.3
Why is the Approximate/Design Problem Hard?
There are several reasons one can give to illustrate the difficulties associated with the Approximate/Design Problem. We now discuss some of these.
First, one must note that the weighted H 0* sensitivity problem, in general, is discontinuous
with respect to plant perturbations, even when the uniform topology on V** is imposed. This has
been demonstrated in [53]. Consequently, simple continuity arguements cannot be used. It must
also be noted, however, that even if it were continuous in the uniform topology, there are many
infinite dimensional plants which cannot be approximated uniformly by real-rational functions (e.g.
a delay; see proposition 2.10.1).
A second difficulty can be attributed to the fact that the associated Hankel operator is often not
compact. When this is the case it cannot be approximated uniformly by finite rank operators, and
again it is not clear how to proceed. How to approximate a non-compact operator then becomes
the non-trivial issue.
Another difficulty can be attributed to the fact that weighted R** optimal solutions generally
exhibit bad properties. More specifically, one can show that the optimal solution is often unbounded
and results in an improper compensator. One can correctly argue that this is usually an existence
issue, nevertheless, it is an issue which a designer must contend with.
The following example illustrates that even uniform approximations can lead to bad results.
Example 6.3.1 (Discontinuity, Instability, "Open-loop Intuition" )
Let our infinite dimensional plant be given by P(s) = ~. Let the weighting function be given by
W = y+1, where 0 < # < 1. The associated optimal compensator is infinite dimensional. It can be
found by solving the infinite dimensional model matching problem defined by:
popt
dtf
I|W(1 -
inf0
QE i
PQ)|. 7.
This has been done in [14]. There, it is shown that
p, = max{ IW(joo)|, |lwe.| } = |Iwe.| |.
Thus, |r|we.1 > 1.
We want to obtain a near-optimal finite dimensional compensator, by solving an appropriately
formulated finite dimensional problem. Let Pa(s) = (.n)n ' define a set of finite dimensional
approximants for P. It can be shown that Pn uniformly aprroximates P on the extended imaginary
axis (cf. example 2.10.1). The approximants Pn are thus terrific, based on "open-loop intuition".
A solution to the finite dimensional optimization problem
Qinf ||W(l- PnQ)II
n
is given by
#) (1+ -)n.
n
This Q, results in pn = |W(joo)| = 1. We note that pi cannot approach Iopt for large n since
pot > 1 = p, for all n E Z+. This has been also noted in [53]. The example thus illustrates the
discontinuity of weighted w* sensitivity measures.
The above Qn generates a finite dimensional compensator
Q= (1
K
-
Q= (# - 1) W(s) (1 + -)".
1 - PnQn
n
Given this, we note that
def
W
W(1-P
Qn)
We also note that
||(Pn - P)Qn||,j
> 1
for all n E Z+. Moreover, it can be shown that Kn does not internally stabilize the plant P for any
value of n E Z+One might argue that this is because Q, is improper to begin with. One might go a step further
to correct the above instability problem by rolling-off Qn with some roll-off function. This can not
be done using a fixed order roll-off function. Even if we were able to guarantee stability, there is
no natural way to modify Qn so that fin is close to popt for large n.
The difficulty lies in the approximants Pn. They are bad approximants of P, given that we are
solving a weighted R** sensitivity problem. They fail to approximate the inner part of the plant.
This is crucial given that we are solving a sensitivity problem.
The above example clearly shows that the approximants P, must be chosen appropriately.
Approximants must be chosen on the basis of a closed loop design objective; not on open loop
intuition. Consequently, which approximants are used is critically dependent on which design
criterion is used.
The example also shows that the way in which we modify the resulting Q,, must be done cleverly.
"Classical" fixed order roll-off functions, in general, will not work.
Finally, the example shows that simple continuity arguments can not be used to obtain a solution
to our Approximate/Design problem.
We now present our solution to the R** Approximate/Design Sensitivity Problem.
6.4
Solution to 7H* Approximate/Design Sensitivity Problem
In this section we shall solve the NX** Approximate/Design Sensitivity Problem. We shall do so by
constructing a near-optimal finite dimensional compensator for the infinite dimensional plant P.
This will be done by appropriately modifying finite dimensional solutions {Qn}** based on the
finite dimensional approximants {Pn}' 1 . The techniques developed in Chapter 5 shall be heavily
exploited.
In this section, the following assumption will be made about the infinite dimensional plant P
and the finite dimensional approximants {Pn}* 1 .
Assumption 6.4.1 (Construction of Approximants and Bezout Factors)
(1) N,, and Dp have a finite number of zeros on the extended imaginary axis; each with finite
algebraic multiplicity.
(2) The sequence {N,.},**L C RX** consists of inner functions which uniformly approximate Np,
on all compact frequency intervals (excluding the point joo); i.e. for each fl E R+, however large,
we have limn,
,
- N)X[_O,nL
= 0.
(3) The sequence {Nn}** C RX** consists of outer functions which uniformly approximate
N,,; i.e. limn.-o ||N,, - Np,||I = 0. Moreover, the sequence is constructed as indicated in
construction 5.5.1.
(4) The sequence {Dn}I**
C RX** consists of inner functions which uniformly approximate
Di; i.e. limn_.o 1Dni - DiIIh0 = 0.
(5) The sequence {D, 0 }*o1 C RX**" consists of outer functions which uniformly approximate
D,; i.e. limn-.,o ||Dn - D,,||
= 0. Moreover, the sequence is constructed as indicated in
construction 5.5.1.
(6) P
d
dfdf
where N,, 1e N,
Nn. and D,, t' DP, Dp.
(7) The sequence {Nk,}**'l C RX-*
uniformly approximates Nk; i.e. imn_.oo |INk, - NkI||.
= 0.
(8) The sequence {Dkr},**
C RN* uniformly approximates Dk; i.e. limeoo IDk, - Dki.
= 0.
Comment 6.4.1 (Applicability, Practicality)
(a) Poles and Zeros on Extended Imaginary Axis.
Condition (1) simply says that P has a finite number of poles and zeros on the extended imaginary axis. Relaxing this condition will be an area for future research.
(b) The Inner Part and Right Half Plane Zeros.
Condition (2) is reasonable since it allows N,1 to be discontinuous at oo. It thus allows for plants
with delays. Delays can be approximated uniformly on compact frequency intervals using Pade'
approximants (cf. example 2.10.2). Such approximants agree with control engineering intuition:
the need to approximate the plant at "low" frequencies.
The condition allows P to have an infinite number of zeros in the open right half plane. P , for
example, may contain an infinite Blaschke product of open right half plane zeros. In such a case,
the partial products can be used as the N,, [30].
If P has an infinite number of open right half plane zeros, then the zeros can only accumulate
at oo. If they were to accumulate within the finite open right half plane, then this would imply
that N, is identically zero in the open right half plane (cf. proposition 2.3.5). If they were to
accumulate on the imaginary axis then N,1 would possess essential singularites at those points and
hence we cold not approximate it uniformly on compact frequency intervals (cf. proposition 2.7.3).
It thus follows that the only point of accumulation can be oo.
(c) Continuity of N,,, D, , and D,.
It should be noted that the approximants in (3), (4), and (5) are guaranteed to exist if and only
if N,,, D,,, D,, E Ce. This follows from proposition 2.10.1.
(d) Approximation of Inner and Outer Parts.
We approximate the inner and outer parts separately, in order to control the pole-zero structure
of the approximants P, on the imaginary axis. If we did not perform the approximations in the
above manner, then the pole-zero structure of P, and P may differ drastically on the imaginary axis,
even for large n. Such a situation is highly undesirable since X** sensitivity solutions are discontinuous with respect to addition of poles and zeros on the imaginary axis. Since the approximants
are based on the construction given in construction 5.5.1, the ideas in construction 5.5.1 are critical.
(e) Right Half Plane Poles.
Condition (4) implicitly forces P to have only a finite number of open right half plane poles.
This follows from proposition 2.7.3.
This makes sense since a plant with an infinite number of open right half plane poles cannot be
stabilized by a finite dimensional compensator.
(f) Continuity of Nk and Dk.
We note that the existence of the approximants in (7) and (8) are guaranteed by assumption
6.2.2 modulo proposition 2.10.1.
Comment 6.4.2 (Stable Plants)
For stable plants we may choose N, = P, N,, = P,
D, = D,, = Dk = Dk, = 1, and Nk=
Nk, = 0. In such a case we choose P,i to approximate P; on compact frequency intervals. We also
choose P,, such that it approximates P, uniformly on the extended imaginary axis. Moreover, Ps,
is choosen, according to construction 5.5.1, so that its zero structure on the imaginary axis does
not differ drastically from that of P0 .
The following lemma shows that if P, is constructed as above, then N,, and D,, will be a
coprime factorization for P, over RX**, with Bezout factors Nk, and Dk..
Lemma 6.4.1 (Algebraic Properties of Approximants)
There exists N E Z+ such that
(D,,Dk, - N,,Nk.)-'
E RR**
for all n > N.
U
Proof
Adding and subtracting appropriately, gives the following equality:
D,,D,-
N,,Nkn, = (D,
-(N,-
NP)(Nk,
- Dp)(Dk,
-
-
Dk) + Dk(DP - D,) + Dp(Dk,
Nk) - Nk(N,
-
N,) -NP(Nk
- N)
-
Dk) + DDk
NNk.
However, by assumption DDk - NNk = 1. The result then follows from assumption 8.4.1.
I
Comment 6.4.3 (Strict Propriety of Nk )
In the above proof we used the strict propriety of Nk along with the fact that N,, approximates
N, uniformly on compact frequency intervals. If Nk were only proper then we would need N,, to
be a uniform approximant, say, in order to guarantee that |INk(Np - N,)|I|
is small for large n.
N
With this lemma and proposition 3.2.1, it follows that the set of all compensators which internally stabilize Pn, with respect to the ring RX**, are parameterized by
K(Pn,Q ) a= Nk, -D,,Q
Dk, -NN,,Q
where Q is any element in RX** [56]. Consequently, the approximants P, as constructed above,
possess the desired algebraic properties.
We now relate the structure of our problem to the finite dimensional K** Model Matching
Problem considered in section 5.5.
Proposition 6.4.1 (Structure)
T,
WD,,Dk,
and
T 2 ln' WD,, N
satisfy the conditions in section 5.5.
Proof
Given the previous lemma, the result follows from assumption 8.4.1. We note, for example, that
= 0
lim ||T1. - TillI7.
and
lim
(T2 , - T2)X[_=,0
0
for each 0 E R+, however large. These follow from the inequalities
IIT1,
-
IIW(D,, - Dp)DknlliI + IIWDp(Dk,
TillW0
-
Dk)|Ii 0
and
!5
(T2, - T2)Xi-n,Ol 1
+
|IIW(Dp, - Dp)NI1,||.
WD,(N,, - N)X[ _
1ou
and assumption 8.4.1.
Other results from section 5.5 will be stated when needed.
The following theorem captures the main ideas in obtaining a solution to the H** Approximate/Design Sensitivity Problem.
Theorem 6.4.1 (Main Ideas)
Suppose that
lim A < pope
(1)
and that there exists a uniformly bounded sequence {Q}n'L=1 C RRO* such that
T,-
T 2 nQ
(2)
pn + E,
for n sufficiently large, and
lim |1 - 6(Pn, kn(Qn))
n-o
7joo
=
0.
(3)
Given the above, {kn}n 1 will internally stabilize P with respect to the ring 00 for all but a
finite number of n. In addition, the actual performance approaches the optimal performance as the
approximantsget "better"; i.e.
lim pn = popt.
Proof
Since
def
W
..
po QEnu-
1 - PK( ,Q)y
and
W
def
(D,
1 - PK(Pn, Qn)
we have It opt
- D,,)(Dk,
- NpQn)
+ (T1n - T 2 nQn)
6(P,kn(Qn)
i
7i
, for each n E Z+. Consequently,
Tn
Iopt
-
+
T 2 nQn
An <_1
-
W(D, - DPn)(Dkn - N,,Qn)
1 - 6(Pn,Kn(On)) 11N.,
The result then follows from conditions (1), (2), and (3) within the theorem.
We now show that (3) implies internal stability. To do so, we argue as follows.
Since Kn internally stabilizes Pn, it follows that Nk, - DPnQn and Dk, - Npn n must be co-
prime in R**. We also have that NP and D,, are coprime in H". Consequently, from proposition
3.3.1, we have that Kn internally stabilizes P if and only if 6(P,kn) is a unit of (i.e. invertible
in) X**. However, (3) implies that 6(P,kn) will be a unit for all but a finite number of n. This
completes the proof.
0
Theorem 6.4.1 shows precisely what is needed to solve the R** Approximate/DesignSensitivity
Problem. It was shown in section 6.3 that one can run into serious difficulty satisfying the third
condition of theorem 6.4.1 if one chooses on = Qn. We emphasize that it is condition (3) which will
allow us to guarantee internal stability when Kn is used with P. Because of this, the "traditional"
choice of r, as the identity, comes with no guarantees. In what follows we shall present a way for
choosing Qn, and hence the roll-off operator r, so that all three conditions in theorem 6.4.1 are
satisfied. With this construction we will have a solution to the N** Approximate/DesignSensitivity
Problem.
We shall see that condition (1) in theorem 6.4.1 will follow from the implicit structure of
the weighted sensitivity problem. This condition should be interpreted loosely as an "uppersemicontinuity" condition. Such a condition should be expected from the results in [53, pp. 345].
Condition (2) in theorem 6.4.1 will be achieved by exploiting the ideas developed in chapter 5.
More specifically, (2) will follow from theorem 5.5.1. We refer to condition (2) as a "sub-optimality"
condition.
Finally, condition (3) in theorem 6.4.1 will follow after showing that the Qn are uniformly
bounded and uniformly roll-off in RN8*. This will be critical in stablishing (3). Since condition (3)
gives us stability, we shall refer to it as the "internal stability" condition.
The main result of this section is now stated in the following theorem.
Theorem 6.4.2 (Main Result: Solution to Approximate/Design Problem)
There exists a sequence {Qn}
There also exists N E Z+ such that
C RXw
p1opt < fn
which is uniformly bounded, uniformly rolls-off.
popt +
E
for all n > N. Consequently, there exists a roll-off operatorr : Q - On E RX*iO such that
lim fn = popt.
Moreover, the sequence {Q }
generates a sequence of finite dimensional, strictly proper, internally stabilizing compensators {kn}wl, where
Kn = Nk, - D,,Qn
D,
-
N,,Qn
and
pape
g<.
1 - PKn
JOO
pt+E
for all n > N. kn is thus nearly-optimalfor all n > N.
Proof
To prove the theorem, we only need to show that conditions (1), (2), and (3) of theorem 6.4.1
can be satisfied.
From proposition 5.5.2, we have that
lim pn ! p1 opt.
This gives us the upper-semicontinuity condition (1) in theorem 6.4.1.
From theorem 5.5.1, there exists a sequence {(Qn}
c RHgo which is uniformly bounded,
uniformly rolls-off. Moreover, there exists N E Z+ such that
Tin - TQn
5 Pn +E
for each n > N. This gives us the sub-optimality condition (2) in theorem 6.4.1.
To complete the proof, we only need to show that condition (3) of theorem 6.4.1 holds; i.e.
lim
n--4oo
1 - b(Pn, kn(On))1
'Hoo
= 0.
To do so we note that
6(Pn, kn(Q,)) = D,(Dk, - N,,Qn) - N,(Nk, - D,,Qn)
=
Dp(Dk, - Dk)
+
DDk - Np(Nk, - Nk) - NNk - Dp(N,, - N,)Qn - N,(D, - D,,)Qn.
Since DDk - NNk = 1, this yields
1 - 6(Pn, kn(On)) = D,(Dk - Dk.) + NP(Nk, - Nk) + D,(N,, - N,)Qn + N,(D, - D,,)On.
Condition (3) of theorem 6.4.1 then follows from assumption 8.4.1.
Given the above, it follows from theorem 6.4.1 that {kn} 1 will internally stabilize P with
respect to the ring 7** for all but a finite number of n. In addition, the actual performance
approaches the optimal performance as the approximants get "better"; i.e.
lim pn = I-opt.
This completes the proof.
Comment 6.4.4 (Internal Stability)
To guarantee internal stability we have used the fact that Nk,, uniformly approximates Nk. If Nk,
approximated Nk uniformly only on compact frequency intervals, then we would need say, N, to
roll-off in order to guarantee that |INp(Nk, - Nk)|Ig is small for large n.
The above theorem shows that given our assumptions on the weighting function, the approximants, and on the plant, there exists a way to construct nearly-optimal, finite dimensional, strictly
proper controllers for the infinite dimensional plant; i.e. in a manner such that the actual performance approaches the optimal performance as the approximants get better. Consequently, we
have solved the R** Approzimate/Design Sensitivity Problem. It should be emphasized that this
has been done by using approximants which converge in the compact topology on V** and do not
necessarily converge in the uniform topology. Moreover, the conditions which the approximants
{Pn}* 1 must satisfy are weak.
Comment 6.4.5 (Direct Construction of Finite Dimensional Compensators)
From theorem 5.2.2, it inunediately follows that we can also construct nearly-optimal infinite
dimensional compensators for P. In principle, one can approximate such compensators to get finite
dimensional compensators. This, however, would defeat our purpose. In our Approximate /Design
approach we intentionally avoid solving infinite dimensional optimization problems.
To perform the above construction would mean that we would have to solve an infinite dimensional "inner problem" for Zpt (cf. proposition 5.2.1). In the context of this work, however, this is
unacceptable.
U
The following examples indicate how the ideas presented in this section can be applied.
Example 6.4.1 (A Stable Plant)
Let our infinite dimensional plant be given by P
NNk = 1 where N, = !,
=
Q.
We then have P = N and DDk
-
D, = 1, Nk = 0, and Dk= 1.
Let the approximants be given by Pn =
, where N,, = Pade(",) and D,, = 1. We then
have D,,Dk, - N,,Nk, = 1 where Nk, = 0 and Dk, = 1. The construction of near-optimal
compensators for P based on Pn, then follows from theorem 6.4.2. Here Pade(n,n) denotes the
[n, n] Pade' approximant for e-' (cf. example 2.10.2).
The resulting nearly-optimal compensator takes on the form
dnef
-Qn
1 - PnQn
where
Q
- ZnjAgB-
Here, IA is an RX"* function which is sufficiently close to fA
A(E, W) E Z+ is sufficiently large and fixed. Also, gB
s+lg,
de
1)
E
E R** where B
*, where A
d'
B(e, A) E Z+ is
sufficiently large and fixed. How hA and gB are precisely constructed, is shown in theorem 5.5.1.
The above, compensator is finite dimensional, strictly proper, and will be nearly-optimal for n
sufficently large. How large n needs to be can be determined from theorem 5.5.1 and the results of
this section.
Example 6.4.2 (An Unstable Plant)
Let our infinite dimensional plant be given by P = }.
and D,
=
Also let P = N
where N, = e
.
If we select the nominal compensator Knom = NkDkj', where Nk = -2e and Dk = '+1-_e 1 then we obtain DDk - NNk = 1. We thus have a coprime factorization (Np, D,) for P over R1**
with Bezout factors (N, Dk).
The problem here is that Nk, and hence Knom, does not roll-off (cf. comment 6.4.3). We need to
construct a nominal compensator Knom 'I NkDk-1 which is strictly proper. Such a compensator
will be found using classical ideas.
Let
We shall obtain the desired compensator based on the "design plant" n.
Knom Lef M
s+ a
(s + b)(s + c)
where a, b, c, M are design parameters. It should be clear that there exists a, b, c, M E R+ such
that Knom(s) internally stabilizes P. To see this one first uses root locus ideas to see that Kn,,m(s)
will stabilize the "design plant" ..
The idea, then is to adjust the parameters so that the "design
loop", determined by .1 and Knom(s), remains stable even in the presence of a loop perturbation
e-'. This can be done by appropriately selecting the gain crossover frequency and the phase margin
of the "design loop". One can choose, for example, a = 1, b = 1000, c = 10b and then adjust M
appropriately.
We now obtain a coprime factorization (Nk, Dk) for Knom over 11 with Bezout factors (N,, D,).
To do so, we first consider the loop determined by P and Knom. It has characteristic "polynomial"
given by
d(s)
(s - 1) (s + b) (s + c) - M e'
(s + a).
By construction, all of its roots lie in the open left half plane. We now define
-
def
Nk = Knom
Dd1
and
4Def~bj
pk
-
,
pk
=
d(s)
+ 1)(s + b)(s + c)
By construction 4k E R'. Also, since d(s) is stable and P is proper, we conclude that 4 is a unit
of Roo. Given this, if we define
def Nk
and
def Dk
Dk=-1
then it follows that DpDk - NNk = 1. Moreover, Nk, Dk E C, and hence by proposition 2.10.1 can
be uniformly approximated by RX** functions. It thus follows that N,, D,, N, and Dk satisfy
the needed conditions.
, where N,, = Pade(nn)
Given the above, we then let our approximants be given by Pn =
Dpn
8+1
D,, =
. Moreover, we let Nk, E R1-o and Dk, E RRX** be approximations of Nk and Dk,
respectively. The construction of near-optimal compensators for P based on Pn, then follows from
theorem 6.4.2. Here Pade(n, n) denotes the [n, n] Pade' approximant for e-' (cf. example 2.10.2).
The resulting nearly-optimal compensator takes on the form
def
where the construction of
previous example.
Qn
Nk, - D,,Qn
Dk, - NPnQn
is performed in a manner which is analogous to that discussed in the
U
6.5
Solution to WOO Purely Finite Dimensional Sensitivity Problem: Computation of Optimal Performance
In practice we often would like to compute or estimate the optimal performancepopt. The following
theorem says that under our assumptions, the expected performance pn approaches the optimal
performance p 1 t.
Theorem 6.5.1 (Solution to Purely Finite Dimensional Problem)
lim pn = popt.
This solves the Purely Finite Dimensional Sensitivity Problem.
Proof
The proof of this follows from that of theorem 5.5.2.
I
It should be emphasized that the above result has been obtained even though the optimal
performance popt need not be continuous with respect to perturbations in the plant P, even when
the uniform topology is imposed [53]. Even if it were continuous, there are many plants in V**
which cannot be approximated by RX' approximants; e.g. a delay (see proposition 2.10.1). Given
this, it is also imperative to point out that the above result has been shown even though the
approximants {Pn}*' need not approximate the plant P uniformly.
The approach usually taken in the literature to compute pop is to solve an infinite dimensional
eigenvalue/eigenfunction problem [14, pp. 28-31], [61), [63, pp. 308]. Theorem 6.5.1, however
implies that in order to estimate popt all we need do is solve a sequence of finite dimensional eigenvalue/eigenvector problems. This computational virtue is better exhibited in the following "spectral" corollary to theorem 6.5.1, which shows explicitly the relationship between the relevant finite
dimensional eigenvalue/eigenvector problems and the infinite dimensional eigenvalue/eigenfunction
problem.
Corollary 6.5.1 (Spectral Implication)
if
max{ |T1(joo)j, max|T1(jWk)| }
,
then
lim
Proof
Trr
=popt.
From proposition 5.3.1, it follows that
IT1
poPt = max{IT1(joo)|, maxIT1(jw)|,
From the assumption, we have p,pt =
5.4.1.
}.
TT; . The result then follows immediately from theorem
Comment 6.5.1 (Hankel Norm Assumption)
The inequality in the above coroallary is satisfied if the points oo, {wk} are essential singularities of
T2. This has been shown in [61]. Without going into detail, this is because in such a case |T1(joo)|
and {|T1(jWk)I} lie in the essential spectrum of the Hankel operator PTT and essential spectra
cannot exceed the operator norm. If, for example, T2 = e-', then the inequality is satisfied.
We note that lemma 5.3.1 may help in choosing T1 so that the Hankel norm assumption in
corollary 6.5.1 is satisfied (cf. [14, chapter 8]).
This corollary shows that given our assumptions, the above norms converge. We emphasize
here that we have proven convergence of the norms even though the finite rank Hankel operators
{ T 1 T }'* do not, in general, converge uniformly to the Hankel operator LT 1 ,T;. This is because
rTT; is usually non-compact and thus cannot be approximated uniformly by a sequence of finite
rank operators [5, pp. 42, 178]. The Hankel operator rTT can be shown to be non-compact,
for example, when the weighting W is only proper and the plant P is a pure delay (T1 = W,
Ty = e-' ). This is seen from Hartman's Theorem on compact Hankel operators (proposition
2.9.2). Consequently, our approach also avoids finding approximate solutions via Hankel operator
approximation theory which falls apart for non-compact operators.
Although the above provides a solution to the Purely Finite DimensionalSensitivity Problem,it
does not really provide insight into the computation of the optimal performanceppt. Such insight
can be obtained from the results in sections 5.4 and 5.5.
Lets assume that P E 7-**. In practice one might compute the approximants Pn using Pade'
approximations [3], for example. Such approximants were presented in example 2.10.2 for the case
where P was a delay.
The following examples illustrate how such approximants can be used to solve problems which
would ordinarily require the solution of an infinite dimensional eigenvalue/eigenfunction problem.
Example 6.5.1 (Tannenbaum)
In this example we consider a problem from [18]. Our infinite dimensional plant is a delay: P = e-'A
100
where A = 1. The weighting function is low-pass and given by W(s) =-I
with a = 100. To
compute |Iwp.|| we first solve
tan(y) + -a= 0
for y, = 1.577137. We then have
|IwP *I = |W(j!A })| 0.00634.
Now let Pn = [n, n] Pade' approximant for P (cf. example 2.10.2). Given this, one can show that
{ T wp,. -
}
{0.00498, 0.00629,0.00634,...
Since IW(joo)| = 0 < |rwp- , we expect imn-.oo jTwp,;.|| = jTwpIj. from theorem 5.4.1. The
computer data shows that we have convergence as predicted. One might argue that this case is
easy since Hankel operator is compact.
Example 6.5.2 (Flamm, Mitter)
In this example we consider a problem from [14], [15]. Our infinite dimensional plant is a delay:
where A = 1. The weighting function is low-pass and given by W(s) = 7 with
P = e#= 0.01. To compute irwp,. we first solve
cot(WA) =
W2
/
for w0 = 0.8676622. We then have
|irwp -|| = |W(jwo)| = 1.5258.
Now let Pn = [n, n] Pade' approximant for P(cf. example 2.10.2). Given this, one can show that
{
wP, . }
=
1.4925,1.5254,1.5258, .. }.
Since IW(joo)I = infwER. IW(jw), it follows form lemma 5.3.1 that |W(joo)I 5 ||Twp-||. Given
this, we expect lim_.oo ||Twp, -|| = ||Twp,.|| from (4) of theorem 5.4.1 and the fact that Pn
uniformly approximates P on compact intervals (see example 2.10.2). Although the Hankel operator
Twp . is non-compact, the computer data shows that we have convergence as predicted.
6.6
Solution to 7
Loop Convergence Sensitivity Problem
Thus far, we have given conditions under which the expected performancepn and the actual performance , approach the optimal performance popt. We now investigate the behavior of the actual
loop shapes. More specifically, we would like to know in what sense, if any, does Qn converge to
a near-optimal solution Q,,t. Since Qn and Qpt are based on "inner solutions" Z and Z,,t, it
makes sense to investigate the convergence of Zn to Zopt. We start with the following assumption.
101
Assumption 6.6.1 (Uniform Convergence)
We shall further assume that
lim T2.. n--+oo
= 0.
.no
Comment 6.6.1 (Applicability)
The above assumption will not be satified for a large class of plants; e.g. P = e-'. This is because
R1* functions are not dense in R**. A delay, for example, cannot be uniformly approximated by
R1X** functions (cf. proposition 2.10.1). The assumption can be satisfied if, for example, T 2j E Ce;
i.e. real-rational (cf. proposition 2.7.3). Weakening the above assumption will be a topic for future
research.
Given the above assumption, we have the following proposition.
Proposition 6.6.1 (Weak* Loop Convergence)
Let Z0,pt E W"* be the unique solution to
mn |IT 1 - T2jZ||,. = |IT 1 - T2jZopt|..
ZEN-O
It then follows that {Z,}*' 1 converges to Zopt in the weak* topology on X**.
Proof
We prove the proposition, we begin by noting that R** is the dual space of the quotient space
£'/h' [30, pp. 137]. Since it is a dual space, its elements may be viewed as bounded linear
functionals acting on the primal space 1/R1. Given this, we recall Alaoglu's theorem which says
that the closed unit ball in a dual space is weak* compact (cf. proposition 2.11.4).
Recall that Zn E RX* is a solution to the finite dimensional inner problem
min
ZERN-
|T ,
11IT1
- T2 Z
=T1 - T2. i Zn I,
.
Since T1 , is uniformly bounded, so is Zn. This is because
||Zn||.
<
zln
-ZER~if
T , - T2.t Z
1
11T,
j0
2|IT1,|iI.
+ ||T1I|I|
From Alaoglu's theorem, it thus follows that Zn possesses a subsequence {Zn(k)}= 1 which converges
in the weak* topology on W' to say L E 'H**. Since T2i(,C/X') C V/I',
this implies that
T1 - T21Zn(k) is weak* convergent to T1 - T2 jL. Proposition 2.11.1 thus gives us the following
inequality:
|IT 1 - T2jL||Ij.
lim IT1 - T2 Z(k),
.
k--+oo\
\
o
From this inequality, we obtain
|IT 1 - T2,L||jH.
lim { IT1n(k) - T2l(k)Znkh4
+ T 1 - T1.(k)
102
+ (T2i - T2 l,))Zn,
}
Using assumption 6.6.1 allows the last two terms to vanish. From proposition 5.2.1 it then follows
that
|IT 1 - T2iL||h.
lim
k-.oo
T, n(k)
1
2
n(k)i
.
From assumption 6.6.1 and theorem 5.4.1 it follows that the right hand side is equal to
From proposition 5.2.1, it thus follows that
|IT 1 - T2 L||y.
T1T;,
=
rTi
T.
|IT 1 - T2zZot||,. = min |IT 1 - T2iZ|l|.-
Since, by assumption, Z,,t is the unique minimum, the above inequality implies that L = Zot.
Moreover, we have also shown that any weak* convergent subsequence of Zn necessarily converges
to Z,t. The result then follows from proposition 2.11.5.
Theorem 6.6.1 (Solution to Loop Convergence Problem)
The sequence {Zn}* 1 converges uniformly to Zpt on all compact frequency intervals.
Proof
The sequence {Zn}**' is uniformly bounded in '**. By the Arzela-Ascoli theorem (cf. proposition 2.12.1), the sequence {Zn}** constitutes a normal family of analytic functions over the
open right half plane. This implies that there exists a subsequence {Zn(k)}* 1 which converges
uniformly to say M on all compact subsets of the open right half plane. From proposition 2.10.2,
it follows that M is analytic in the open right half plane. Since Zn is uniformly bounded in H**, it
then follows that M E H0*. Using limiting arguments, one can show that uniform convergence on
compact subsets of the open right half plane implies uniform convergence on all compact frequency
intervals of the imaginary axis.
Lets view Zn(k) and M as bounded linear functionals acting on the primal space C1 /1. Using
the uniform convergence on compact frequency intervals and the fact that functions in C /X1
roll-off, gives us that Zn(k) has weak* limit M. From proposition 6.6.1, it follows that M = Z,t.
Consequently, Zn(k) converges to Z,,t on compact frequency intervals.
The above shows that any compactly convergent subsequence {Z(k))" 1 must converge to Z,,t.
Suppose that Zn does not converge compactly to Z,,. Then, there exists a subsequence
{Zn(k)1**1 and a compact set S such that
(Zn(k) - Zot)Xs
>
for all k. This, however, contradicts the normality of {Zn}*
verges to Zot uniformly on all compact frequency intervals.
E
. It must therefore be that Z, con-
Comment 6.6.2 (Convergence of Loop Shapes)
The above theorem implies that if proper care is taken, we can construct sequences of compensators
which yield loop shapes which approach the optimal loop shape uniformly on compact subsets. The
practical implications of this are obvious. This makes the Approximate/Design approach presented
in this chapter a truely useful design tool.
This concludes our discussion of the H** Loop Convergence Sensitivity Problem.
103
6.7
Summary
In this chapter a solution was presented to the R'*Approximate/Design Sensitivity Problem. More
specifically, it was shown how near-optimal finite dimensional strictly proper compensators could be
constructed for an infinite dimensional plant, given a weighted RX1* sensitivity design specification.
It was shown that the construction could be carried out on the basis of finite dimensional solutions
obtained from appropriately formulated finite dimensional NX** sensitivity problems. The finite
dimensional problems which one solves are "natural" weighted R** sensitivity problems obtained by
replacing the infinite dimensional plant by "appropriately" chosen finite dimensional approximants.
It was shown that, in general, approximants based on "open loop intuition", rather than on
the control objective, may yield compensators which do not even guarantee stability when used
with the infinite dimensional plant. It was also shown how "appropriate" approximants could be
constructed. The approximants obtained were constructed so that their imaginary axis pole-zero
structure would not drastically differ from that of the plant. The construction presented does not
require sophisticated mathematics or software. It can be used by practicing engineers with little
effort.
We also provided a solution to the Purely Finite Dimensional RO* Sensitivity Problem. Here,
the issue of computing the optimal performance was addressed. It was shown that the optimal
performance could be computed by solving a sequence of finite dimensional eigenvalue/eigenvector
problems rather than the typical infinite dimensional eigenvalue/eigenfunction problems which
appear in the literature. This makes a once difficult problem, almost trivial. Examples were given
to illustrate this.
Finally, conditions were given under which the near-optimal finite dimensional loop shapes could
converge to the optimal infinite dimensional loop shape uniformly on compact frequency intervals.
This was illustrated in our solution to the R** Loop Convergence Sensitivity Problem. This makes
the Approximate/Design approach presented in the chapter a truely promising engineering tool.
104
Chapter 7
Design via 7oo Mixed-Sensitivity
Optimization
7.1
Introduction
In this chapter we consider the problem of designing near-optimal finite dimensional compensators
for infinite dimensional plants via VX" mixed-sensitivity optimization. Such an approach can be
motivated by design specifications which require some specified degree of robustness or L2 disturbance rejection. A systematic procedure is presented. More specifcally, we provide a solution
to the W"* Approximate/Design Mixed-Sensitivity Problem, the R-*0 Purely Finite Dimensional
Mixed-Sensitivity Problem, and the H1*0 Loop Convergence Mixed-Sensitivity Problem. Since the
ideas are similar to those presented in the previous chapter, we shall focus on stable plants.
7.2
'
Approximate/Design Mixed-Sensitivity Problem
In this section we present some definitions and assumptions to precisely state the Hi* Approximate/Design Mixed-Sensitivity Problem. Notation to be used throughout the chapter is also established.
Since we assume that our infinite dimensional plant is stable we have that P E RO*. Given this,
it follows from proposition 3.2.1 that the set of all compensators which internally stabilize P, with
respect to the ring H*, are parameterized by
K(P,Q)-1-PQ
1 -PQ
where Q is any element in w".From this, it follows that if we allow Q to vary over X-a", then we
get all strictly proper compensators which internally stabilize P. We shall be doing this throughout
the chapter; i.e. all infimizations involving P shall be carried out over HO.
In this chapter we shall consrtruct R1** approximants {P}**1 for P. From proposition 3.2.1,
it follows that the set of all compensators which internally stabilize Pn, with respect to the ring
R**, are parameterized by
K(Pn, Q))e
1
-PnQ
where Q is any element in 1*. From this, it follows that if we allow Q to vary over ?YW, then we get
all strictly proper compensators which internally stabilize Pn. We shall be doing this throughout
the chapter; i.e. all infimizations involving Pn shall be carried out over XYw.
105
MEN==
In this section we shall formulate an HX mixed-sensitivity problem. To do so, we shall need
two frequency dependent weighting functions, W and W 2 . The following "standard" assumption,
on the weighting functions W 1 and W 2 , shall be made throughout the chapter.
Assumption 7.2.1 (Weighting Functions)
(1) W1 E RX"0 and minimum phase.
(2) W 2 ,W2 £ RER.
The above implies that W 1 and W 2 have no poles or zeros in the closed right half plane. W may
or may not roll-off.
Suppose our mixed-sensitivity criterion is based a performance measure
J(PK(P Q)) -
()
=
1 - PK(P,Q)W2Q-
(Wl(1
PQ
7)
which penalizes the sensitivity and complementary sensitivity transfer functions. Here Q varies
over 1O" since we are interested only in strictly proper compensators. It is easy to see that such a
criterion possesses pathologies similar to those possessed by the R" sensitivity problem. We note,
for example, that such a criterion will do its best to invert the outer part of P. The criterion,
will thus, in general generate improper compensators. Although this difficulty, as will become
apparent, can be addressed using the techniques presented in the previous chapters, it can be
altogether avoided.
In this chapter we shall present new results for a mixed-sensitivity criterion which, in our
opinion, is a "better" and more "natural" criterion. The criterion we shall consider is one which
penalizes the transfer function associated with the control, as well as the sensitivity function. By
doing so we will be able to obtain proper compensators to start with. Obtaining a suboptimal
finite dimensional compensator which rolls-off then becomes the central issue. We shall see that to
achieve this objective, all one need do is apply the ideas presented in Chapter 5 for the sensitivity
problem. To provide a logical flow of ideas we proceed as we did for the sensitivity problem in
Chapter 6.
To begin our development, we define the X" mixed-sensitivity measure to be used throughout
the chapter as follows.
Definition 7.2.1 (X"O Mixed-Sensitivity Measure)
Let F, G, Q G R1". Also let K(G, Q) denote a compensator which internally stabilizes G with
respect to the ring W"'. If it also internally stabilizes F, it is appropriateto define the R' mixedsensitivity measure of the pair (F, K(G, Q)) as follows:
J(FK(G,Q))
1- FK(GQ)
- G 00
106
From proposition 3.2.1 it follows that K(G, Q) =
. We thus have
WI(1 - GQ))
J(FK(GQ)) =
W2Q
\
1-(G - F)Q
The above shows that when II(F - G)QI|,. < 1, then K(G, Q) "internally" stabilizes F and J
is well defined. This follows from the Small Gain Theorem [11]. We also see that if F = G, then
this "small gain" assumption is automatically satisfied, and hence J is well defined. This is because
K(F,Q) "internally"stabilizes F for any Q E hm, by definition. These issues shall receive further
consideration below.
From definition 7.2.1 above, it follows that J(P, K(P,Q)) =
W(i - PQ))
. Substituting into definition 5.2.1 and allowing
Q to vary over
\
W2Q
)
innoo
RH", then gives us the following expression for
the optimal performancepupt.
Definition 7.2.2 (Optimal Performance)
def
(W 2 K(Pq, Q))
0Lpr 1 - PK(P,Q)
inf
\
Eur
WQ P
70
Q
17foo
We emphasize that this definition defines an infinite dimensional optimization problem; one which
we want to, and will, avoid solving. We denote its infimizer by Qpt eE 1"1.
Similarly, from definition 7.2.1 above, it follows that J(P,K(P,Q)) =
After substituting this into definition 5.2.1, and allowing
following expression for the expected performance p,,.
Q
W(i - PQ))
W2Q
hoo.
to vary over R i", one obtains the
Definition 7.2.3 (Expected Performance)
(W 2 K(P, Q))
der
QeRuL
Q ERWO
fW
in
1 - PnK(PnQ)
11h7oo
1 (1-PnQ)
W 2Q
QERu0
01
/
2
Here we can infimize over R2o" since W, W 2 , and P are real-rational [23].
Let Q, achieve the infimum (or be near-optimal) in definition 8.2.3. We note that in general
Q, will lie in Xw. This is because W 2 is invertible in ?". Consequently, Q, will generate a finite
dimensional compensator
K
def
=-
-Qn
K(Pn, Qn) = 1 -
"PQ
which generally does not roll-off. Also Kn will not necessarily stabilize the infinite dimensional
plant P (cf. example 7.3.1).
'Existence issue ???
107
In an attempt to remedy these shortcomings, we define a roll-off operator
r : Q,-+Q
E R~e**
The exact form of r, as we shall see, is obtained in a manner analogous to that used for the
sensitivity problem. To construct r all that will be required are the ideas in theorem 5.2.1. The
exact form of r will be determined subsequently.
The compensator generated by On is given by
Kn
K(P,Q,)= 1
Given this, we consider the feedback system obtained by substituting kn into a closed loop
system with the infinite dimensional plant P. Let H(P,kn) denote the resulting closed loop
tranfser function matrix from r, d to e, u. We then have
[e
H(Pkn)
where
H(PlIn)
=
[
"g
,
P1
1
-
1 -PR,
-
1.
1-PR,
J
Substituting for kn, then gives
H(P,kn(Qn)) -
1 - PnQn P(1 - PnQn)1
-On
1 - PnQn
1 -(Pn - P)Q*
Given that internal stability can be shown, the actual performancej~ defined in definition 4.3.3
is well defined and becomes:
Definition 7.2.4 (Actual Performance)
def
(W(1W2Qn
\W2K(Pnn))
S1-
PK(PnQO)
1- (Pn -P)Q
)
'
Given the above definitions, the H* Approximate/Design Mixed-Sensitivity Problem then becomes to find conditions on the approximants {Pn}* 1 , and on the roll-off operator r, such that
the actual performance approaches the optimal performance; i.e.
lim pn = ptp
Equivalently, this problem can be viewed as that of finding a near-optimal compensator for the
infinite dimensional plant P. The problem also addresses the question: What is a "good" finite
dimensional approximant?
Because this problem is of primary concern in this chapter, we now indicate what difficulties
are associated with the problem.
108
7.3
Why is the Approximate/Design Problem Hard ?
The idea behind the Approximate/Designapproach is that if P is close to P, then on should be
close to 1 0,p. One also might expect yn to be close to yop. This is not true. The following example
shows that even in simple situations we may have a discontinuity.
Example 7.3.1 (Discontinuity)
Let P =
be our plant. Suppose
def
Povt =
in
mf
(W
2
PP,
IIW2||ue.
< |W1(joo)|. Also suppose that
Q)
=inf
1 - PK(P,Q)
QEoo
QENoo
W (1 - PQ)
W 2 PQ
00
Here we penalize the sensitivity and complementary sensitivity functions. We now show that pot
is discontinuous at P. We first note that
popt >! |W1( j
Let Pa =
g-
oo|
> ||W2||uhx .
approximate P. We see that P approximates P uniformly in R**. Also let
W1+
1W2PnK(PnQ
- PnK(Pn,Q)
Sin
GGRo
We see that choosing
Q
=
-
E HX*
QERuN
W1
PnQ
W2PnQ)
x('
shows that
pn 5 ||W2||u.- < |W1(joo)| 5 Ypt-e
Consequently, yn cannot approach popt, as n approaches infinity, even though Pn does approach
P. We thus have a discontinuity at P.
The above example shows that when the sensitivity and complementary sensitivity are penalized,
then the resulting optimization problem exhibits pathologies identical to those possessed by the 1**
sensitivity problem. It should be noted, however, that the techniques developed in Chapter 6 can
be used to overcome these pathologies. Rather than taking this approach, we choose to use a nicer
cost function; namely one which penalizes the control to reference transfer function rather than
the complementary sensitivity transfer function. When this is done, the discontinuity exhibited in
the above example does not occur. This has been shown in [53] under the uniform topology on
R*. We shall show that useful results can be obtained when working with the compact topology
on H**.
We now present our solution to the R** Approximate/Design Mixed-Sensitivity Problem.
109
7.4
Solution to 71 Approximate/Design Mixed-Sensitivity Problem
In this section we shall solve th ** Approximate/Design Mixed-Sensitivity Problem. We shall do
so by constructing a near-optimal finite dimensional compensator for the infinite dimensional plant
P. This will be done by appropriately modifying finite dimensional solutions {Qn}l* based on
finite dimensional approximants {Pn}**. The techniques developed in Chapter 5 shall be heavily
exploited.
In this section, the following "compact convergence" assumption shall be made on the finite
dimensional approximants {Pn}n*.
Assumption 7.4.1 (Construction of Approximants)
The sequence {P,1}** E RV** is uniformly bounded in R** and uniformly approximates P on all
compact frequency intervals; i.e. for each f E R+, however large, we have
= 0.
limn (Pn - P)X[_O,n
Comment 7.4.1 (Practicality)
Compact approximants are easily obtained. If P is a delay, for example, one can use Pade' approximants (cf. example 2.10.2 ). Other approximants can also be used (cf. example 2.10.1 and
example 2.10.3).
The following theorem captures the main ideas in obtaining a solution to the V'
mate/Design Mized-Sensitivity Problem. The theorem is analogous to theorem 6.4.1.
Approxi-
Theorem 7.4.1 (Main Ideas)
Suppose that
lim pn
opt,
(1)
and that there exists a uniformly bounded sequence {Qn}n0
W1(1 - PnQn)
W2Qn
C RXOO such that
n+E(2
</pn+±E,
(2)
1H 0
for each n sufficiently large, and
lim |(Pn - P)QnO
n--oo |
\Ho
=0.
Then kn will "internally" stabilize P for all but a finite number of n and
lim Yn = plopt.
110
(3)
Proof
The proof of this theorem is identical to the proof of theorem 6.4.1.
Theorem 7.4.1 shows precisely what is needed to solve the H1-* Approximate/Design MixedSensitivity Problem. Comments analogous to those given for theorem 6.4.1 can be made here.
The following theorem shows how one might construct a strictly proper nearly-optimal compensator for the infinite dimensional plant P. We note that such a compensator is guaranteed to
exist by the definition of the optimal performancepopt in definition 7.2.2. We present the theorem
because it will provide insight into the subsequent construction of r, Qn, and hence Kn.
Throughout the section, we shall require the following roll-off function.
Definition 7.4.1 (Roll-off Function)
(1
def
h
where m E Z+.
Such a function was discussed in Chapter 2 and heavily exploited in Chapters 5 and chapsensol.
Theorem 7.4.2 (Near-Optimal Irrational Compensator)
There exists M 1
Mi (e, ppt) E Z+ such that
(W1(l-QPQ,,thMl)
W2QopthM1
If we define Q.Pt
- Qopthm1 , then Qopt
0
E
< popt + 3E
)
~10
H 'o and we have that
W1(1W- PQopt)
))
P
W2
o
Qopt
0oo
1
po
pp0t ± 3e.
E
Given this,
-def
Kopt
-0opt
1 - PQ0pt
will be a a strictly proper nearly-optimal compensatorfor P.
I
Proof
Since the proof of this will be insightful, it shall be given. We have that Qo,,pt f Qopthm where
Qopt E H" achieves the infimui in definition 4.3.1. By definition Qopt E R** and
W1 (1 -
2PQopt)
W20opt
2
IIuo
* sup IW1(1 - PQopthm)12 + IW2 Qopthm12
>o
To prove the result, it will suffice to show that
(W(1
- PQopthm)) 2
)
W2Qopthm
111
<,P2
O 5opt
+
for m sufficiently large. To do this we first define
def W~
Y =' W 1 (1
Q~~
PQopt),
-
T1 jel W1,
and
T =' T1(joo).
Given this, by adding and subtracting one gets the following
W1(1 - PQopthm) = (TI - T)(1 - hm) + T + (Y - T)hm.
This then implies that
W1(1 - PQpthm)
W2Qopthm
2
)
,h
sup{I(T1 - T)(1 - hm)1 +21(TI - T)(1 - hm)|T +(Y - T)hm+ T + (Y - T)hm| 2 + IW2 Qopthm| 2 }
2
w>O
From this we obtain
W1(1 - PQopthm))
W2Qopthm
2
< II(T1 - T)(1 - hm)|112.
W
WO
+ 211(T1 - T)(1 - hm)(T + (Y - T)hm)II|.
+ sup{IT + (Y - T)hm| 2 + W2Qopthm| 2 }.
w>O
Since T + (Y - T)hm is uniformly bounded in 'H* and since hm approximates unity on compact
frequency intervals, from lemma 2.10.1 it follows that
(W(1
- PQopthm))
W2Qopthm
2
) 7j
! 2 + sup { IT + (Y - T)hm| 2 + IW2Qopthm| 2 }
w>O
for m sufficently large. To prove the result we thus only need to consider the last two terms.
Using the algebraic result in lemma 5.2.2, we know that given e > 0, m can be chosen such that
IT + (Y - T)hmI2 < |T12 (1 - 2|hm| + IhmI 2) + IYI 2Ihm| 2 + 2|TIIYIIhmI(1 - ihm|) + E
almost everywhere on the imaginary axis. This is the key step. It follows because
(1) |TI and IY| are uniformly bounded,
(2) IIhmjI||.
< 1, and
(3) hm has special phase properties.
This, then implies that
IT+(Y-T)hm| 2 +W 2Qopt| 2 < IT| 2 (1-2|hm|+|hm|2 )+2|TI|YIhm|(l-hm|)+(Y| 2+W 2Qopt| 2)|hm|
almost everywhere on the imaginary axis.
From proposition 2.7.2, it follows that
|TI
popt.
112
2 +E
We also note that
IY| 5 popt.
By definition, Q0 pt satisfies
|Y| 2 + IW2Qt| 2
(
(
OP
W1(1 - PQopt))
W2Qopt
)
2
u.j
almost everywhere on the imaginary axis. Combining the above inequalities then gives
IT + (Y - T)hmI|2 + IW2 Qopthm| 2
<
2t(1
- 2|hmI + Ihm| 2 + 2|hml - 21hm1 2 ) + ipthm| 2 +
E
almost everywhere on the imaginary axis. This then implies that
sup
w>O
{
|T + (Y - T)hm| 2 + IW2 Qopthm| 2
1,2+
E
for m sufficently large. Given this, it then follows that m can be chosen such that
(Wi(1
- PQthm))
)u
\ W200,thm
2
,
2
3c.
e
This proves the result.
Comment 7.4.2 (Main Ideas)
The main ideas in the above proof are as follows:
(1)
pt is finite,
(2) IIhmII|.u
1,
(3) the phase properties of hm.
These ideas shall be used shortly to get a result analogous to theorem 7.4.2.
We now present the following "upper-semicontinuity" result for the mixed sensitivity problem.
It gives us condition (1) in theorem 7.4.1.
Lemma 7.4.1 (Upper-Semicontinuity)
lim pn
popt.
U
Proof
From definition 7.2.2 for popt, we know that given e > 0 there exists Qo E 7R'
W(1 -
QO)LOpt+E.
7j h
\etW2
We then have
An=in
Ego||
1
(1 - Pn Q) )| 11
W2Q
|--
113
f W11W2Q0PQo))I
00u
such that
|| W1(1 - PQo)\
1||
W2Q0
) 11
+IW(n-PQjU
||
||0
1(,-H0Q
which after using the near optimality of Qo gives
< 1A+ E+ ||W 1 (Pn - P)QII| 0
Finally, by assumption 7.4.1 on the approximants, the form of Q, , and lemma 2.10.1 we can make
the right term arbitrarily small by taking n sufficiently large. Since e can be arbitrarily small this
proves the result.
Comment 7.4.3 (Uniform Boundedness)
Lemma 7.4.1 implies that pn is a uniformly bounded sequence. This fact is crucial in the proof of
the following lemma.
U
The following lemma is analogous to lemma ?? and gives us the "sub-optimality" condition (2)
in theorem 7.4.1.
Lemma 7.4.2 (Near Sub-Optimality)
There exists
f
E R(o such that
W1(1 - PnQnf))
.
|\ W2Qnf
/
+
p+,
C,
for n sufficiently large. Moreover, if we define a roll-off operatorr : Qn On
n
E RhO where
e Qnf,
then
W1(1 - PnQn)
W2n
for each n
&p
Y
+C
Wo0
E Z+. Given this, kn will be a finite dimensional strictly proper near-optimalcompen-
sator for Pn for each n c Z+.
Proof
The proof is identical to that given for theorem 7.4.2. The "upper-semicontinuity" result
of 7.4.1 guarantees that pn will be bounded. This is critical in the proof. See comment 7.4.2. Here
one can use f def
= Lim where hm is simply an RX** function which is sufficiently "close" to the irrational function hm in the X** topology. Such a function is guaranteed to exist by lemma 2.10.1. U
Given the above, we can obtain the "stability" condition (3) in theorem 7.4.1.
Lemma 7.4.3 (Internal Stability)
The sequence {Qn}**=
1 C R H j* is uniformly bounded and uniformly rolls-off in w.
limn |(Pn - P)Q,
114
= 0.
Moreover,
Proof
0 we have the following:
Since W 2 is invertible in X"
||Qnl||i
! jW
||j.jW2Qn||wH
(WW1(
W
PnQn)
<jW
pjj,.n.
PpOt (cf. lemma 7.4.1), we can conclude that Qn is uniformly bounded in
pn
Since limn
-o. The result then follows from the form of Qn, assumption 7.4.1 on the approximants, and
I
lemma 2.10.1 which allows us to turn compact convergence into uniform convergence.
We now state the main result of this chapter which guarantees the existence of a near-optimal
strictly proper finite dimensional compensator.
Theorem 7.4.3 (Main Result: Solution to Approximate/Design Problem)
There exists a roll-off operatorr : Q --+ Qn E RHIo such that
lim pn:= popt.
Moreover, the
Q,
constructed in lemma 7.4.2 define such an operator.
Proof
The proof follows from theorem 7.4.1 which captures the main ideas, the results in lemmas 7.4.1
7.4.2, and 7.4.3 and the construction of the sequence {Q}n. 1 .
This solves the Nw Approximate/Design Mixed-Sensitivity Problem.
7.5
Solution to 7110 Purely Finite Dimensional Mixed-Sensitivity
Problem: Computation of Optimal Performace
1
In practice we often would like to compute or estimate the optimal performance, opt. The following
theorem says that the expected performance,y actually approaches the optimal performance,popt.
Theorem 7.5.1 (Solution to Purely Finite Dimensional Problem)
lim pn = Iopt.
Proof
We have already shown in lemma 7.4.1 that
lim A 5 popt.
I
The proof of the converse inequality parallels the proof given in theorem 6.5.1.
This solves the Nw Purely Finite Dimensional Mixed-Sensitivity Problem.
Usually the optimal performance iopt is computed by solving an infinite dimensional eigenvalue/eigenfunction problem involving a Hankel - Toeplitz operator pair [40]-[41], [61]. The above
theorem shows that popt can be computed by solving a sequence of finite dimensional problems.
As in corollary 6.5.1, one can associate finite dimensional eigenvalue/eigenvector problems with
these finite dimensional problems. Consequently, to estimate pot, we only need to solve a sequence
of finite dimensional eigenvalue/eigenvector problems. Moreover, the ideas of [12] can be used to
assist with numerical computations.
115
7.6
Poles and Zeros on the Imaginary Axis
Unlike with the Xw sensitivity problem, the mixed-sensitivity problem does not suffer from imaginary axis pole-zero discontinuities.
7.7
Solution to 7-f
lem
Loop Convergence Mixed-Sensitivity Prob-
Thus far, we have given conditions under which the expected performace p, and the actual performance pn approach the optimal performance p,pt. Results analagous to those in section 6.6 can be
obtained which guarantee the convergence of the actual loop shapes. Moreover, it is not necessary
that Pn approximate P uniformly.
7.8
Unstable Plants
Unstable plants can be treated similarly. Additional care must be taken to guarantee stability.
This is done by approximating the factors Nk, Dk, D, uniformly. N, need only be approximated
on compact frequency intervals.
7.9
General Weighting Functions
The theory presented in this and the previous chapter can be generalized to handle more general
weights. Weights in RX" offer a designer a considerable amount of flexibility. Such an extension
will not be pursued in this work.
7.10
Super-Optimal Performance Criteria
Often, we might like to optimize a given performance criterion subject to an additional constraint.
For example, Xw sensitivity minimization subject to an X" or an L bound. Given the ability
to obtain proper finite dimensional compensators for finite dimensional versions of such problems,
the techniques presented thus far can be used to construct strictly proper finite dimensional nearoptimal compensators for the infinite dimensional plant. Such performance criteria are well-posed
in the sense of [53]; i.e. they are continuous with respect to plant perturbations in the uniform
topology on H".
7.11
Summary
In this chapter solutions were presented to the X" Approzimate/Design , Purely Finite Dimensional, and Loop Convergence Mixed-Sensitivity Problems.
116
Chapter 8
Design via
8.1
2 Optimization
Introduction
In this chapter we consider the problem of designing near-optimal finite dimensional compensators
for infinite dimensional plants via 7X2 optimization. Such an approach can be motivated by design specifications which require some specified degree of robustness or 2 disturbance rejection.
A systematic procedure is presented. More specifcally, we provide a solution to the N 2 Approximate/Design Sensitivity Problem, the 7 2 Purely Finite Dimensional Sensitivity Problem, and the
72 Loop Convergence Sensitivity Problem. Solutions to the corresponding Mixed-sensitivity problems shall also be disussed. Again, we focus on stable plants in order to isolate the key ideas.
8.2
W2 Approximate/Design Sensitivity Problem
In this section we present some definitions and assumptions to precisely state the R2 Approximate/Design Sensitivity Problem. Notation to be used throughout the chapter is also established.
Since we shall assume that our infinite dimensional plant is stable we have P E 'XH". Given
this, it follows from proposition 3.2.1 that the set of all compensators which internally stabilize P,
with respect to the ring H1, are parameterized by
K(P,Q)
-PQ
1 -PQ
where Q is any element in H 0 . From this, it follows that if we allow Q to vary over *11, then we
get all strictly proper compensators which internally stabilize P. We shall be doing this throughout
the chapter; i.e. all infimizations involving P shall be carried out over RO*
In this chapter we shall construct RX** approximants {P}**
1 for P. From proposition 3.2.1,
it follows that the set of all compensators which internally stabilize Pn, with respect to the ring
R**, are parameterized by
1
-
Q
where Q is any element in w*. From this, it follows that if we allow Q to vary over RO, then we get
all strictly proper compensators which internally stabilize Pn. We shall be doing this throughout
the chapter; i.e. all infimizations involving Pn shall be carried out over HO*.
In this section we shall formulate an 72 weighted sensitivity problem. To do so, we shall require
a frequency dependent weighting function W. The following assumption shall be made on W.
117
Assumption 8.2.1 (Weighting Function)
(1) W E Rh 2.
(2) W is outer.
(3) W(s) =
"' where ck E C and Re(ak)
<
0.
We now define the notion of an R 2 -sensitivity measure as follows.
Definition 8.2.1 (W" -Sensitivity Measure)
Let Q E Roo and F, G, E R00. Also, let K(G, Q) denote a compensator which internally stabilizes
G with respect to the ring hw. If it also internally stabilizes F, it is appropriate to define the
H2 -sensitivity measure of the pair (F, K(G,Q)) as follows:
def
W
J 1 - FK(G, Q)
7J2
. Substituting
From definition 8.2.1 above, it follows that J,2 (P, K(P,Q))
into definition 4.3.1 and allowing Q to vary over 7g, then gives us the following expression for the
optimal performance, popt.
Definition 8.2.2 (Optimal Performance)
inf
1 - PK(P,Q) H2
NQ
p Cle
=
inf ||W(1 - PQ)II 2
QENu
We emphasize that this definition defines an infinite dimensional optimization problem; one which
we want to, and will, avoid solving. Moreover, we note that the problem is an 2 Model Matching
Problem. It is analogous to the R'* problem studied in section 5.2.
Q) I . After
Similarly, from definition 8.2.1, it follows that Ju 2 (Pn, K(P,, Q))
substituting this into definition 5.5.1, and allowing Q to vary over RRW, one obtains the following
expression for the expected performance, pn.
Definition 8.2.3 (Expected Performance)
def.Win
yn "
inf
9Ru%
-
1- P=K(PnQ)
P0(nQ
u
inf
Gu
IW(-
||W(1 - PnQ)|I,.
Here, we can infimize over R1w since W and Pn are real-rational. We note that this definition
defines a sequence of finite dimensional model matching problems.
An optimal or near-optimal solution to this problem is typically found by first considering the
"inner problem":
inf
||W - PniZ||u 2 -
Here, Pnj is the inner part of P.
118
From the classicalprojection theorem, we know that there exists Zn E RXi*
||W - PniZn||u, = zmin
such that
||W -Pni Z||uJ2.
Moreover, Zn is unique and is given by
Zn= IlsW
P*
Lets define
Qn
=
W-'P- Zn
where Pn, is the outer part of P. This Qn generates a finite dimensional compensator
K
t'= K(Pn, Qn) =
1
PnQn
.
This compensator, as we expected, need not even stabilize P. We thus need to modify it. For this
reason, we define a roll-off operator
r : Qn -_+On ERR'**
The exact form of r will be determined subsequently.
The compensator generated by Qn is given by
kn'-K(P,
n
Qn) =
1
- PnQn
.
Given this, we consider the feedback system obtained by substituting Rn into a closed loop
system with the infinite dimensional plant P. Let H(P,kn) denote the resulting closed loop
transfer function matrix from r, d to e, u. We then have
e H(P,kn)
,
where
H(P,kn)
]
1~;kn
1-Pk-
1-PRn
1-PRn
Substituting for kf, then gives
H(PI
n(On))
[-P~
[ -On
n
P(-Pnn)
1 - Pn~n
J1
-(Pn
P)Qn*
Given that internal stability can be shown, the actual performance, 4n defined in definition
4.3.3 is well defined and becomes:
Definition 8.2.4 (Actual Performance)
def
-1
W
-Pkn
-
7J2
119
W(1 - PQn)
1 -(Pn - P)Qn u2
Given the above definitions, the 7j 2 Approximate/Design Sensitivity Problem then becomes to
find conditions on the approximants {P,}*' 1 , and on the roll-off operatorr, such that the actual
performance approaches the optimal performance; i.e.
limfA
= popt
Equivalently, this problem can be viewed as that of finding a near-optimal compensator for the
infinite dimensional plant P. The problem also addresses the question: What is a "good" finite
dimensional approximant?
Because this problem is of primary concern in this research, we now indicate what difficulties
are associated with the problem.
Why is the Approximate/Design Problem Hard?
8.3
There are several reasons one can give to illustrate the difficulties associated with the Approximate/Design Problem. We now discuss some of these.
First, one must note that the weighted H 2 sensitivity problem, in general, is discontinuous
with respect to plant perturbations, even when the uniform topology on R** is imposed. This has
been demonstrated in [53]. Consequently, simple continuity arguments cannot be used to obtain a
solution to our Approximate/Design Problem. It must also be noted, however, that even if it were
continuous in the uniform topology, there are many infinite dimensional plants which cannot be
approximated uniformly by real-rational functions (e.g. a delay; see proposition 2.10.1).
Another difficulty can be attributed to the fact that weighted R 2 optimal solutions generally
exhibit bad properties. More specifically, one can show that the optimal solution is often unbounded
and results in an improper compensator. One can correctly argue that this is usually an existence
issue, nevertheless, it is an issue which a designer must contend with.
The following example illustrates that even uniform approximations can lead to bad results [53].
Example 8.3.1 (Discontinuity of
2
R
Sensitivity Problem)
Let our infinite dimensional plant be given by P(s) = Q. Let the weighting function be given
The associated optimal compensator can be found by solving the infinite dimensional
by W = ,.
model matching problem defined by:
def
p
=,p
inf ||W(1 - PQ)||w .
Using classical projection theory, one obtains
s
+1
(1
-
s+1
=
Q
7
1
es
1 aQ) 2
-
s+1
> 1+
2Q
+
s+1s+1
-1
e-W
s+ 1
1
(S + 1) 2 Q
l,
1
2
'We-'
7,2
120
= (1
+ e~ 2).
_-
(+
-
Q-
2
The last term can be made arbitrarily small by appropriate choice of
Lopt =
__
s +
2
2
e
2
-_
=
(s + 1)2
e 2)7r +
e- 1
es
2
Q.
Consequently,
1)2
We want to obtain a near-optimal finite dimensional compensator, by solving an appropriately
formulated finite dimensional problem. Let Pa(s) = (n)n _1T define a set of finite dimensional
approximants for P. It can be shown that Pn uniformly aprroximates P on the extended imaginary
axis (cf. example 2.10.1). The approximants P are thus terrific, based on "open-loop intuition".
However, simple analysis shows that pn = II 2 -WP*
= 0 < popp for each n E Z+. We thus
have a discontinuity; i.e. pn does not approach p,,p for large n, even though P approaches P.
Although the approximants selected appeal to our open loop intuition, it is clear that they do
not help in achieving the closed loop objective. The approximants selected are bad because they fail
to approximate the inner part of the plant. Doing so is important when solving an NX2 sensitivity
problem.
The above example clearly shows that the approximants Pn must be chosen appropriately.
Approximants must be chosen on the basis of a closed loop design objective; not on the basis of
open loop intuition. Consequently, which approximants are used is critically dependent on which
design criterion is used.
Finally, the example shows that simple continuity arguments can not be used to obtain a solution
to our Approximate/Design problem.
We now present our solution to the 2 Approximate/Design Sensitivity Problem.
8.4
Solution to '2
Approximate/Design Sensitivity Problem
In this section we shall solve the H 2 Approximate/Design Sensitivity Problem. We shall do so by
constructing a near-optimal finite dimensional compensator for the infinite dimensional plant P.
This will be done by appropriately modifying finite dimensional solutions {Qn}** based on the
finite dimensional approximants {P}* 1 . The techniques developed in Chapter 5 shall be heavily
exploited.
In this section, the following assumption will be made about the infinite dimensional plant P
and the finite dimensional approximants {P}**l.
Assumption 8.4.1 (Construction of Approximants and Bezout Factors)
(1) P has a finite number of zeros on the extended imaginary axis; each with finite algebraic
multiplicity.
(2) The sequence {P }** C RW"* consists of inner functions which uniformly approximate P on
all compact frequency intervals excluding the point joo); i.e. for each 0 E R+, however large, we
have limno (Pa, - Pi)X[n,O]
= 0.
(3) The sequence {P,,}**1 C R7i* consists of outer functions which uniformly approximate P,;
i.e. i _ ||P, - P0 1,n,. = 0. Moreover, the sequence is constructed as indicated in
construction 5.5.1.
121
Comment 8.4.1 (Applicability, Practicality)
(a) Zeros on Extended Imaginary Axis.
Relaxing condition (1) will be an area for future research.
(b) Inner Part and Right Half Plane Zeros.
Condition (2) is reasonable since it allows Np, to be discontinuous at oo. It thus allows for plants
with delays. Delays can be approximated uniformly on compact frequency intervals using Pade'
approximants (cf. example 2.10.2). Such approximants agree with control engineering intuition:
the need to approximate the plant at "low" frequencies.
The condition allows P to have an infinite number of zeros in the open right half plane. P , for
example, may contain an infinite Blaschke product of open right half plane zeros provided that the
zeros accumulate at oo only. In such a case, the partial products can be used as the N, [30].
If P has an infinite number of open right half plane zeros, then the zeros can only accumulate on
the imaginary axis or at oo. If they were to accumulate within the finite open right half plane, then
this would imply that N, is identically zero in the open right half plane (cf. proposition 2.3.5).
If they were to accumulate on the imaginary axis then Pi would possess essential singularities at
those points and hence we could not approximate it uniformly on compact frequency intervals (cf.
proposition 2.7.3). It thus follows that the only point of accumulation can be oo.
(c) Continuity of P,.
It should be noted that the approximants in (3) are guaranteed to exist if and only if P, c Ce.
This follows from proposition 2.10.1.
(d) Approximation of Inner and Outer Parts.
We approximate the inner and outer parts separately, in order to control the pole-zero structure
of the approximants P, on the imaginary axis. If we did not perform the approximations in the
above manner, then the pole-zero structure of P, and P may differ drastically on the imaginary
axis, even for large n. Such a situation is highly undesirable since it would complicate the inversion
of P.. We would like to invert P,, as we invert P. To do so, they must possess similar imaginary
zero structures. This was seen in Chapter 6. Since the approximants are based on the construction
given in construction 5.5.1, the ideas in construction 5.5.1 are critical.
(e) Compact Approximants.
Finally , we note that (2) and (3) imply that the sequence {Pa},*o 1 C R-** uniformly approximates P on all compact frequency intervals (excluding the point joo); i.e. for each 0 E R+,
however large, we have lim-.o 1(Pn - P)X[O,nd1
= 0.
From proposition 3.2.1, it follows that the set of all compensators which internally stabilize Pn,
122
with respect to the ring RR**, are parameterized by
K (Pn ,Q)
~tc
1-PQ
where Q is any element in RX** [56]. Consequently, the approximants Pn, as constructed above,
possess the desired algebraic properties.
The following theorem captures the main ideas in obtaining a solution to the R 2 Approximate/Design Sensitivity Problem.
Theorem 8.4.1 (Main Ideas)
Suppose that
lim pn < popt
(1)
and that there exists a uniformly bounded sequence {Qn}n 1 c RXw such that
JW(1 - PnQn,2 < pn + f,
(2)
for n suffilciently large, and
lim
(P,
P
-
=
0.
(3)
Given the above, {k},**=1 will internally stabilize P with respect to the ring RO* for all but a
finite number of n. In addition, the actual performance approaches the optimal performance as the
approximantsget "better"; i.e.
lim f, = popt.
Proof
Since
W
def
QE0 1-PK(P,Q)
7J2
and
-
def
W
W(1
1 - PK(PnjQn)
we have popt
72
PnQn)
1 - (Pn - P)Q,
'2
[In for each n E Z+. Consequently,
5 n 5
Pot
JW(1
PnQn,
2
1 - I(Pn - P)Qnj, O
The result then follows from conditions (1), (2), and (3) within the theorem.
We now show that (3) implies internal stability. To do so, we argue as follows. Since kn internally stabilizes Pn, it follows that -Q and 1 - PnQn must be coprime in R**. We also have that
Pn and 1 are coprime in W1**. Consequently, from proposition 3.3.1, we have that Kn internally
stabilizes P if and only if 1 - (Pn - P)Qn is a unit of (i.e. invertible in) R**. However, (3) implies that 1 - (Pn - P)Qn will be a unit for all but a finite number of n. This completes the proof. I
Theorem 8.4.1 shows precisely what is needed to solve the 'H2 Approximate/Design Sensitivity
Problem. It was shown in section 6.3 that one can run into serious difficulty satisfying the third
123
condition of theorem 8.4.1 if one chooses 0. = Qn. We emphasize that it is condition (3) which will
allow us to guarantee internal stability when k. is used with P. Because of this, the "traditional"
choice of r, as the identity, comes with no guarantees. In what follows we shall present a way for
choosing on, and hence the roll-off operator r, so that all three conditions in theorem 8.4.1 are
satisfied. With this construction we will have a solution to the H 2 Approximate/Design Sensitivity
Problem.
We shall see that condition (1) in theorem 8.4.1 will follow from the implicit structure of
the weighted sensitivity problem. This condition should be interpreted loosely as an "uppersemicontinuity" condition. Such a condition should be expected from the results in [53, pp. 345].
Condition (2) in theorem 8.4.1 will be achieved by exploiting the ideas developed in Chapter 5.
We refer to condition (2) as a "sub-optimality" condition.
Finally, condition (3) in theorem 8.4.1 will follow after showing that the Q, are uniformly
bounded and uniformly roll-off in R11'O. This will be critical in stablishing (3). Since condition (3)
gives us stability, we shall refer to it as the "internal stability" condition.
The main result of this section is now stated in the following theorem.
Theorem 8.4.2 (Main Result: Solution to Approximate/Design Problem)
There exists a sequence {0n}0 1 C RH'
There also exists N E Z+ such that
which is uniformly bounded, uniformly rolls-off.
popt < An
popt
+
C
for all n > N. Consequently, there exists a roll-off operatorr : Qn -+ On E RNO such that
lim en = popt.
Moreover, the sequence {Q}n 1 generates a sequence of finite dimensional, strictly proper, internally stabilizing compensators {kn}n=1 , where
1 -
and
Itopt
PnQn
W
1
PKn
u
opt
for all n > N. kn is thus nearly-optimal for all n > N.
Proof
To prove the theorem, we only need to show that conditions (1), (2), and (3) of theorem 8.4.1
can be satisfied. We proceed in six steps.
Step 1: Upper-semicontinuity Condition.
Let Q0 E ** be such that pot
IIW(1 - PQ0 )II 2 + c. The upper-semicontinuity condition
then follows from the following inequality
pn
IIW(1 - PnQO)||u 2
IIW(1 - PQo)||U2 + IIW(Pn - P)QO)||uj.
124
Step 2: Uniform Boundedness and Uniform Roll-off.
Given that Zn = Iu2 WP*i
,
we note that
Zn = E
I
c.
k=1
P* (-ak) .
s + ak
It thus follows that Zn is uniformly bounded and uniformly rolls-off in H1o0 .
Let n def ZnW-'P'g where g E R1*. We know from Chapter 5 that g can be chosen such
that W- 1 Pgig is uniformly bounded in N**. Thus Qn is uniformly bounded and uniformly rolls-off
in H**.
Step 3: Computation of pn.
Simple analysis shows that pAn = IH2 W
72
=W
-
Pn, Zn||2.
Step 4: Sub-optimality Condition.
We note that
JW(l - PQ
7,2
- PZn||K
2 + ||Zn(1 -
-|W
g)I|2-
The first term is pA. The second term can be made arbitrarily small by choosing g appropriately.
That such a function g exists follows from Chapter 5. We thus have
Pn~n)1
W(1-
<
pn
+E
for all n E Z+. This gives the sub-optimality condition (2).
Step 5: Internal Stability Condition.
To complete the proof, we only need to show that condition (3) of theorem 8.4.1 holds; i.e.
lim (Pn - P)Q1 'Hoo = 0.
n-+#oo
This, however, follows since Q uniformly rolls-off and since Pn approximates P uniformly on
compact frequency intervals (cf. lemma 2.10.2).
Step 6: Conclusion.
Given the above, it follows from theorem 8.4.1 that {k}n* 1 will internally stabilize P with
respect to the ring H** for all but a finite number of n. In addition, the actual performance
approaches the optimal performanceas the approximants get "better"; i.e.
lim pn = popt.
oo
h-
This completes the proof.
125
The above theorem shows that given our assumptions on the weighting function, the approximants, and on the plant, there exists a way to construct nearly-optimal, finite dimensional, strictly
proper controllers for the infinite dimensional plant; i.e. in a manner such that the actual performance approaches the optimal performance as the approximants get better. Consequently, we
have solved the ? 2 Approximate/Design Sensitivity Problem. It should be emphasized that this
has been done by using approximants which converge in the compact topology on R** and do not
necessarily converge in the uniform topology. Moreover, the conditions which the approximants
{P,-} must satisfy are weak.
Comment 8.4.2 (Direct Construction of Finite Dimensional Compensators)
From theorem 5.2.2, it immediately follows that we can also construct nearly-optimal infinite dimensional compensators for P. In principle, one can approximate such compensators to get finite
dimensional compensators. This, however, would defeat our purpose. In our Approximate /Design
approach we intentionally avoid solving infinite dimensional optimization problems.
To perform the above construction would mean that we would have to solve an infinite dimensional "inner problem" for Ztt (cf. proposition 5.2.1). In the context of this work, however, this is
unacceptable.
U
8.5
Solution to 'H2 Purely Finite Dimensional Sensitivity Problem: Computation of Optimal Performance
In practice we often would like to compute or estimate the optimal performance popt. The following
theorem says that under our assumptions, the expected performance pn approaches the optimal
performance pLpt.
Theorem 8.5.1 (Solution to Purely Finite Dimensional Problem)
lim pn = popt.
This solves the X 2 Purely Finite Dimensional Sensitivity Problem.
Proof
An upper-semicontinuity result was established in the proof of theorem ??. To prove
this theorem it suffices to prove a lower-semicontinuity result. The proof of this follows from the
following inequality
popt
where
Q is selected
W(1 - PQn)2
<
W(1 - PnQn)
j.2
+ IW(Pn - P)Qn)
7j2
U
as in theorem 8.4.2.
It should be emphasized that the above result has been obtained even though the optimal
performance popt need not be continuous with respect to perturbations in the plant P, even when
the uniform topology is imposed [53]. Even if it were continuous, there are many plants in R**
which cannot be approximated by RX* approximants; e.g. a delay (see proposition 2.10.1). Given
this, it is also imperative to point out that the above result has been shown even though the
approximants {Pn},* need not approximate the plant P uniformly.
126
The following corollary gives great insight into the computation of the optimal performance
opt .
Corollary 8.5.1 (Computation of Optimal Performance)
lim
Proof
11
H2_ W P
= pop.
First we note that
Popt =
H
2
LWP
7.j2
This follows from simple analysis. The proof then follows from the inequality
|1 U
-LW P*
-
U2WPi*
I|
<2
U2W
(Pn
-
P)
*
I
and lemma 2.10.2.
Comment 8.5.1 (Implication)
The above corollary implies that the optimal performance pot can be computed by solving a
sequence of Lyapunov equations. We recall that if F(s) = C(sI - A)-1B E RH 2 , then
||F|7
2 =
trace{B'MB}
where M is the unique symmetric solution to the Lyapunov equation
A'M + MA + C'C = 0.
M is the observability grammian of F.
U
8.6
Solution to 7j2 Loop Convergence Sensitivity Problem
Thus far, we have given conditions under which the expected performance pi and the actual performance f, approach the optimal performance yut. We now investigate the behavior of the actual
loop shapes. More specifically, we would like to know in what sense, if any, does on converge to
a near-optimal solution Qcpt. Since on and Q,, are based on "inner solutions" Zn and Z0 pt, it
makes sense to investigate the convergence of Zn to Zopt. We start with the following assumption.
Theorem 8.6.1 (Solution to Loop Convergence Problem)
If Zopt E V** is the solution to
min |W - PiZ||jy. = ||W - PZoptI| |.7 ,
ZEW-
then
lim ||Zn - Zopt||I.
fl--+oo
127
= 0.
__
__
U
Proof
First we note that
Z,,
Zn
IW
Zand = I
ZOPt
H
H'HWP,*iP*i
,*.=1
ca
17sW P * =
I c
= I: Ck
2W
k=1
(- ak)
ki
Pi*(--a )
(ak)
8+ak
The result then follows since the ak lie in a compact set and P,
compact sets.
uniformly approximates P on
Comment 8.6.1 (Convergence of Loop Shapes)
The above theorem implies that if proper care is taken, we can construct sequences of compensators which yield loop shapes which approach the optimal loop shape uniformly. The practical
implications of this are obvious. This makes the Approximate/Design approach presented in this
chapter a truely useful design tool.
This concludes our discussion of the H 2 Loop Convergence Sensitivity Problem.
8.7
Unstable Plants
In this section we exploited the fact that W was real rational. If P is unstable then we need to deal
with additional factors D, and Dk and N,. How to deal with these additional terms is a topic for
future research.
8.8
Mixed-Sensitivity and Super-Optimal Performance Criteria
Extension of the ideas to mixed-sensitivity and super-optimal performance criteria is straight forward.
8.9
Summary
In this chapter a solution was presented to the H2 Approximate/Design Sensitivity Problem. More
specifically, it was shown how near-optimal finite dimensional strictly proper compensators could be
constructed for an infinite dimensional plant, given a weighted H 2 sensitivity design specification.
It was shown that the construction could be carried out on the basis of finite dimensional solutions
obtained from appropriately formulated finite dimensional R 2 sensitivity problems. The finite
dimensional problems which one solves are "natural" weighted 1t 2 sensitivity problems obtained by
replacing the infinite dimensional plant by "appropriately" chosen finite dimensional approximants.
It was shown that, in general, approximants based on "open loop intuition", rather than on
the control objective, may yield compensators which do not even guarantee stability when used
with the infinite dimensional plant. It was also shown how "appropriate" approximants could be
constructed. The approximants obtained were constructed so that their imaginary axis pole-zero
128
structure would not drastically differ from that of the plant. The construction presented does not
require sophisticated mathematics or software. It can be used by practicing engineers with little
effort.
We also provided a solution to the Purely Finite Dimensional R2 Sensitivity Problem. Here,
the issue of computing the optimal performance was addressed. It was shown that the optimal
performance could be computed by solving a sequence of finite dimensional eigenvalue/eigenvector
problems rather than the typical infinite dimensional eigenvalue/eigenfunction problems which
appear in the literature. This makes a once difficult problem, almost trivial. Examples were given
to illustrate this.
Finally, conditions were given under which the near-optimal finite dimensional loop shapes could
converge to the optimal infinite dimensional loop shape uniformly on compact frequency intervals.
This was illustrated in our solution to the R 2 Loop Convergence Sensitivity Problem. This makes
the Approximate/Design approach presented in the chapter a truely promising engineering tool.
129
Chapter 9
Summary and Directions for Future
Research
9.1
Summary
In this thesis three new problems were formulated. They were the (1) K-Norm Approximate/Design
J-Problem, the (2) K-Norm Purely Finite Dimensional J-Problem, and the (3) K-Norm Loop
Convergence J-Problem. Solutions were presented for X" and H 2 sensitivity and mixed-sensitivity
performance criterion. We now summarize the results obtained for each problem and criterion.
It has been shown how one can systematically design near-optimal finite dimensional compensators for infinite dimensional plants, based on finite dimensional approximants. The criteria
used to determine optimality are standard weighted I" and ?X2 sensitivity and mixed-sensitivity
performance measures.
More specifically, it has been shown that given an "appropriate" finite dimensional approximant
for an infinite dimensional plant, one can solve a "natural" finite dimensional problem and modify the solution, using appropriate "roll-off" functions, in order to obtain a near-optimal, strictly
proper, finite dimensional compensator. The "natural" problem is obtained by substituting the
finite dimensional approximants in place of the infinite dimensional plant in the original optimization problem. The term "appropriate" was made precise and depended on the closed loop design
objective.
For the -"' and H2 sensitivity problems, "appropriate" approximants were constructed by
approximating the inner and outer parts of the plant separately. The inner part was approximated
uniformly on compact frequency intervals and the outer part was approximated uniformly. The
approximants of the outer part were constructed such that their zero structure did not differ
drastically from that of the plant on the extended imaginary axis. By doing so, the inversion of
the outer part of the approximants could be done uniformly and in a bounded manner. Such an
inversion was necessary to insure stability. The need to approximate the inner and outer parts
separately is tied to the fact that the optimal performance, when using a sensitivity performance
criterion, is intimately dependent on the inner part of the plant. Such a criterion always require
the inversion of the outer part.
For the Xw and H 2 mixed-sensitivity problems, in which control action is penalized, it was
shown that approximating the plant directly on compact sets would yield "appropriate" approximants. We emphasize that such approximants can be computed from frequency domain data.
This summarizes our results for the K-Norm Approximate/Design J-Problemfor X-" and H2
sensitivity and mixed-sensitivity criterion.
130
In addition, we showed that for the H, problems, the optimal performance can be computed
by solving a sequence of finite dimensional eigenvalue/eigenvector problems rather than the typical
infinite dimensional eigenvalue/eigenfunction problem which appear in the literature. This could
be done even in situations where the corresponding Hankel-Toeplitz operators were non-compact.
For the H2 problem, the optimal performance could be computed by solving sequences of
finite dimensional Lyapunov equations in order to compute the 2 norms of real-rational functions
associated with the approximants.
This summarizes our results for the K-Norm Purely Finite DimensionalJ-Problemfor '" and
H2 sensitivity and mixed-sensitivity criterion.
For each performance criterion considered, conditions were given under which loop convergence
could be guaranteed. That is, conditions under which the actual loop shapes approach the optimal
loop shapes in a strong sense.
This summarizes our results for the K-Norm Loop Convergence J-Problem for X" and X2
sensitivity and mixed-sensitivity criterion.
In summary, our approach allows us to forgo solving a "complex" infinite dimensional HX"0 and
H2 problems. It provides rigorous justification for some of the approximations that control engineers
typically make in practice. In addition, it has been clearly demonstrated that approximants must
be chosen on the basis of the closed loop control objective and not on the basis of "open loop"
intuition.
9.2
Directions for Future Research
Obvious directions for future research include extensions to Ll performance criterion and to multivariable infinite dimensional systems. Such extensions are currently in progress.
In our sensitivity results, the inner and outer parts of the plant were approximates separately.
Inner-outer factorizations may be difficult to obtain in practice. We would like to obtain conditions
under which only the plant P must be approximated. This would enable us to generate designs on
the basis of frequency response data.
Also associated with sensitivity paradigms are issues which arise when an infinite number of
poles and zeros are present. Such issues must be carefully studied and understood.
Since our methods revolve around finite dimensional approximants, approximation algorithms
must be developed, convergence rate experiments must be conducted, and covergence rate results
need to be obtainded. These issues are currently being addressed by many researchers in both the
mathematics and engineering disciplines.
Finally, we contend that the ideas presented extend to areas such as system identification and
distribute control. What you make small in an identification algorithm should depend on the control
objective; not on some stability proof. What knowlege local controllers should have depends on
the global closed loop system objective.
131
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